MGU Mathematics Series — Calculus

Part V: Applications of Integration, Differential Equations, and Series

Sequences · Series · Geometric · Power Series · Taylor Polynomials

Chapter 14 of 14

MGU Mathematics Series · Calculus · Library Textbook Edition

Chapter 14:
Sequences, Series,
and Calculus Readiness
for Higher Mathematics

Infinite processes, convergence and divergence, geometric series, power series, Taylor polynomials, error estimation, and the bridge to Calculus II

■ Math Level: Calculus I ■

■ 14 Sections • Part V of 5 ■

Follows Chapters 1–13 · Prepares students for Calculus II, Differential Equations, and Higher Mathematics

Chapter 14 — Front Matter

Purpose and Overview

This chapter introduces the ideas of sequences and series and builds readiness for the deeper treatment those topics receive in Calculus II. It is designed as a bridge, not a complete theory course. Students who have worked through the previous thirteen chapters have learned how calculus describes change and accumulation through limits, derivatives, and integrals. This chapter now asks a different kind of question: what happens when a process goes on forever? What does it mean to add infinitely many numbers? Can an infinite sum of terms have a finite value? Can a function be expressed as an infinite polynomial?

These questions are not tricks or theoretical curiosities. They sit at the heart of how mathematics, physics, engineering, computing, and data science describe the world. Taylor polynomials allow calculators to compute trigonometric and exponential values. Power series allow physicists to solve differential equations that resist closed-form solutions. Geometric series allow financial analysts to price annuities and perpetuities. Fourier series allow engineers to analyze signals, music, and heat flow.

This chapter builds the conceptual vocabulary and intuition students need before encountering the full theory of convergence in Calculus II. Every idea introduced here is genuine. Every warning about what is being simplified for now is honest. Students who finish this chapter should feel prepared — not overwhelmed — for what lies ahead.

Opening Question

A calculator can display the value of $\sin(0.3)$ almost instantly. But how does it actually compute that? There is no simple algebraic formula for $\sin(0.3)$. The calculator is not looking up a table. It is computing an approximation using a polynomial — a finite piece of an infinite sum. The formula behind this computation is:

$\displaystyle\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$

This chapter begins the process of understanding what that formula means, where it comes from, and how it represents a function as an infinite sum of terms.

14Sections

11Objectives

22Key Terms

10+Worked Examples

Learning Objectives

1Define a sequence and identify its terms from a formula or a pattern.
2Describe what it means for a sequence to converge or diverge.
3Define an infinite series as the limit of partial sums.
4Recognize and evaluate geometric series.
5Determine whether a geometric series converges or diverges, and find its sum when it does.
6Identify the structure of a power series and describe its role as a generalization of polynomials.
7Write and evaluate Taylor polynomial approximations for functions near a point.
8Estimate the error in a Taylor polynomial approximation using the Lagrange remainder.
9Connect series ideas to applications in physics, computing, and engineering.
10Recognize common misunderstandings about sequences and series.
11Articulate readiness for the deeper convergence theory of Calculus II.

Sections at a Glance

14.1Why Infinite Processes MatterThe half-distance paradox; infinite sums producing finite results; central insight of series theory
14.2Sequences ReviewTerms, index, explicit and recursive formulas; graphing sequences; examples of growth, decay, and oscillation
14.3Convergence and Divergence as IdeasLimit of a sequence; convergent and divergent examples; monotone and bounded sequences
14.4Infinite SeriesSeries as sums of sequences; sigma notation review; meaning of an infinite sum
14.5Partial SumsDefinition of partial sums; convergence as a limit of partial sums; telescoping series
14.6Geometric SeriesCommon ratio; partial sum formula; convergence theorem; financial applications
14.7Tests for Convergence as a PreviewDivergence test; harmonic series; integral test, comparison test, ratio test, p-series preview
14.8Power Series as an IntroductionDefinition; radius of convergence; geometric power series; interval of convergence
14.9Functions as Infinite PolynomialsSeries for $e^x$, $\sin x$, $\cos x$; visual comparison with partial sums; polynomial arithmetic on functions
14.10Taylor Polynomials as Local ApproximationsTaylor polynomial formula; Maclaurin polynomials; why the formula works; examples for $e^x$, $\sin x$, $\ln x$
14.11Error and ApproximationRemainder; Lagrange error bound; estimating error; choosing how many terms to use
14.12Series in Physics, Computing, and EngineeringCalculator computation; Fourier series; small-angle approximation; numerical methods; financial mathematics
14.13Common Series MisunderstandingsSeven major misconceptions; harmonic series warning; rearrangement; interval of convergence limits
14.14Readiness for Calculus II and Higher MathematicsWhat Calculus I has built; what Calculus II adds; pathways beyond; readiness checklist

Section 14.1

Why Infinite Processes Matter

Human intuition is built around finite things. We count a finite number of objects, travel a finite distance, wait a finite amount of time. The idea that infinitely many steps could produce a meaningful, finite result can feel impossible at first. But infinite processes are not exotic mathematics invented for its own sake — they appear naturally in many practical situations.

Consider the following scenario. You are standing two meters from a wall. You walk halfway to the wall. Now you walk halfway again. Now halfway again. Each step covers half the remaining distance. Your steps cover distances of 1 meter, then 0.5 meter, then 0.25 meter, then 0.125 meter, and so on without end. Mathematically, you are taking infinitely many steps. Yet physically, you reach the wall. The total distance is:

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 2

An infinite process can produce a finite answer. This is the central insight of series theory. Not every infinite sum converges to a finite value — some grow without bound — but the ones that do converge are extraordinarily useful. Understanding when an infinite process settles down and when it does not is the main goal of series study in Calculus II. This chapter builds the intuition and vocabulary for that study.

Key Principle

Infinite does not mean large. It means the process continues without stopping. Whether it produces something finite depends on how fast the terms shrink.

Section 14.2

Sequences Review

A sequence is an ordered list of numbers. The numbers in the list are called terms. The position of a term in the list is given by an index, usually denoted $n$. A sequence can be written as $a_1, a_2, a_3, a_4, \ldots$ or more compactly as $\{a_n\}$. The index $n$ typically begins at 1 or 0, depending on context.

Explicit Formulas

An explicit formula defines each term directly in terms of its index $n$.

Formula	First Five Terms	Behavior
$a_n = \dfrac{1}{n}$	$1,\ \tfrac{1}{2},\ \tfrac{1}{3},\ \tfrac{1}{4},\ \tfrac{1}{5}$	Decreasing toward 0
$a_n = 2^n$	$2,\ 4,\ 8,\ 16,\ 32$	Growing without bound
$a_n = (-1)^n$	$-1,\ 1,\ -1,\ 1,\ -1$	Oscillating, never settling
$a_n = \dfrac{n}{n+1}$	$\tfrac{1}{2},\ \tfrac{2}{3},\ \tfrac{3}{4},\ \tfrac{4}{5},\ \tfrac{5}{6}$	Increasing toward 1

Recursive Formulas

A recursive formula defines each term in relation to the terms before it. The Fibonacci sequence is a classic example: $a_1 = 1,\ a_2 = 1,\ a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$, giving $1, 1, 2, 3, 5, 8, 13, 21, \ldots$

Graphing a Sequence

A sequence is a function whose domain is the positive integers. Rather than a smooth curve, its graph consists of isolated points at positions $(1, a_1), (2, a_2), (3, a_3), \ldots$

Graph instruction: Plot the sequence $a_n = \frac{1}{n}$ at points $(1,1), (2, 0.5), (3, 0.333), (4, 0.25), \ldots$ The points approach the horizontal axis from above, suggesting convergence to 0.

Section 14.3

Convergence and Divergence as Ideas

The most important question about a sequence is: does it settle down to a single value as $n$ grows? If the terms of a sequence approach a single real number $L$ as the index $n$ grows without bound, the sequence converges to $L$:

\lim_{n \to \infty} a_n = L

If no such number $L$ exists — because the terms grow without bound, oscillate forever, or behave irregularly — the sequence diverges.

Sequence	Converges / Diverges	Reason
$\dfrac{1}{n}$	Converges to 0	Terms decrease to 0
$\dfrac{n}{n+1}$	Converges to 1	Limit of ratio = 1
$n$	Diverges	Terms grow without bound
$(-1)^n$	Diverges	Oscillates between ±1
$\ln(n)$	Diverges	$\ln n \to \infty$

Monotone Bounded Convergence

Every sequence that is both monotone (entirely increasing or entirely decreasing) and bounded (kept between fixed upper and lower limits) must converge. If the terms are always moving in one direction but are constrained within limits, they must eventually pile up near some value. This theorem will be proved rigorously in Calculus II.

Example 14.3.1

Identifying convergence from a formula

ProblemDetermine whether $a_n = \dfrac{3n^2 + 1}{2n^2 - 5}$ converges or diverges. If it converges, find the limit.
ThinkThis is a rational expression in $n$. Divide numerator and denominator by the highest power of $n$, which is $n^2$.
Calculate Divide by n²: lim (3 + 1/n²) / (2 − 5/n²) As n → ∞, 1/n² → 0 and 5/n² → 0 = (3 + 0) / (2 − 0) = 3/2
CheckThe leading terms of numerator and denominator are both degree 2 with leading coefficients 3 and 2. Their ratio is $\frac{3}{2}$. ✓
InterpretThe sequence converges to $\frac{3}{2}$. As the index grows, the terms get closer and closer to 1.5 and stay there.

Answer: The sequence converges. $\displaystyle\lim_{n\to\infty} a_n = \frac{3}{2}$

Section 14.4

Infinite Series

A series is what you get when you add the terms of a sequence. An infinite series is the sum of all the terms of an infinite sequence:

\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + a_4 + \cdots

We cannot literally add infinitely many numbers in finite time. So the value of an infinite series is defined through a limit, using the concept of partial sums developed in Section 14.5.

Sigma Notation

The symbol $\Sigma$ (sigma) denotes a sum. The notation $\displaystyle\sum_{n=1}^{k} a_n$ means: start with $n = 1$, compute each $a_n$, and add all terms through $n = k$.

Notation	Expanded Form	Value
$\displaystyle\sum_{n=1}^{4} \frac{1}{n}$	$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$	$\approx 2.083$
$\displaystyle\sum_{n=0}^{3} 2^n$	$1 + 2 + 4 + 8$	$15$
$\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2}$	$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$	$\dfrac{\pi^2}{6}$ (famous result)

Section 14.5

Partial Sums

The formal definition of the value of an infinite series uses partial sums. The $k$-th partial sum of the series $\sum_{n=1}^{\infty} a_n$ is the finite sum of the first $k$ terms:

S_k = \sum_{n=1}^{k} a_n = a_1 + a_2 + \cdots + a_k

Definition of Series Convergence

The infinite series $\displaystyle\sum_{n=1}^{\infty} a_n$ converges to a sum $S$ if the sequence of partial sums converges to $S$:

$\displaystyle\lim_{k \to \infty} S_k = S$

If the sequence of partial sums diverges, the series diverges and has no finite sum.

Example 14.5.1

Partial sums of a telescoping series

ProblemConsider $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$. Compute partial sums and determine whether the series converges.
ThinkApply partial fractions: $\dfrac{1}{n(n+1)} = \dfrac{1}{n} - \dfrac{1}{n+1}$. This is a telescoping series — most terms cancel.
Calculate S₁ = (1 − 1/2) = 1/2 S₂ = (1 − 1/2) + (1/2 − 1/3) = 1 − 1/3 = 2/3 S₃ = 1 − 1/4 = 3/4 S₄ = 1 − 1/5 = 4/5 In general: Sₖ = 1 − 1/(k+1) = k/(k+1)
Check$\displaystyle\lim_{k\to\infty} \frac{k}{k+1} = 1$ ✓
InterpretThe series converges to 1. The partial sums approach 1 from below, getting closer and closer without reaching it in any finite step.

Answer: The series converges. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1$

Section 14.6

Geometric Series

The most important and accessible family of infinite series is the geometric series. Each term is obtained from the previous one by multiplication by a fixed constant called the common ratio $r$. The general geometric series starting from $n = 0$ is:

\sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \cdots

Geometric Series Convergence Theorem

$\displaystyle\sum_{n=0}^{\infty} ar^n = \dfrac{a}{1-r} \quad \text{if and only if } |r| < 1$

If $|r| \geq 1$, the series diverges. The key is the size of the ratio, not its sign.

Why the Formula Works

Write $S_k = a + ar + \cdots + ar^k$. Multiply both sides by $r$ to get $rS_k = ar + ar^2 + \cdots + ar^{k+1}$. Subtract: $S_k - rS_k = a - ar^{k+1}$, so $S_k = a\cdot\dfrac{1 - r^{k+1}}{1-r}$. When $|r| < 1$, $r^{k+1} \to 0$ as $k \to \infty$, leaving $S = \dfrac{a}{1-r}$.

Example 14.6.1

Evaluating a convergent geometric series

ProblemFind the sum of $\displaystyle\sum_{n=0}^{\infty} 3 \cdot \left(\tfrac{1}{4}\right)^n$.
ThinkIdentify $a = 3$, $r = \tfrac{1}{4}$. Since $\left|\tfrac{1}{4}\right| = 0.25 < 1$, the series converges.
Calculate S = a / (1 − r) = 3 / (1 − 1/4) = 3 / (3/4) = 3 · (4/3) = 4
CheckFirst terms: $3 + 0.75 + 0.1875 + \cdots$. Sum already approaching 4 quickly. ✓
InterpretEven though there are infinitely many terms, their combined contribution is exactly 4.

Answer: $\displaystyle\sum_{n=0}^{\infty} 3 \cdot \left(\tfrac{1}{4}\right)^n = 4$

Example 14.6.2

A divergent geometric series

ProblemDoes $\displaystyle\sum_{n=0}^{\infty} 5 \cdot 2^n$ converge?
ThinkHere $a = 5$, $r = 2$. Since $|r| = 2 > 1$, the series diverges.
InterpretEach term is larger than the one before it. The partial sums grow without bound. There is no finite sum.

Answer: The series diverges. No finite sum exists.

Example 14.6.3

A financial application — present value of a perpetuity

SituationA savings account pays 0.4% interest per month. You deposit $100 at the start of every month indefinitely. What is the present value of all those deposits?
ThinkThe present value of a deposit made at month $n$ is $\dfrac{100}{(1.004)^n}$. The total present value is an infinite geometric series with $a = 100$, $r = \dfrac{1}{1.004} \approx 0.99602$.
Calculate PV = 100 / (1 − 1/1.004) = 100 / (0.004/1.004) = 100 · (1.004/0.004) = 100.4 / 0.004 = $25,100
InterpretThe present value of receiving $100 per month forever at 0.4% monthly interest is $25,100. Geometric series are fundamental to financial mathematics.

Answer: Present value = $25,100

Section 14.7

Tests for Convergence as a Preview

Calculus II includes a full toolkit for determining whether a series converges or diverges. This section previews the most important tests so students are not encountering them cold. The goal here is recognition, not mastery.

The Divergence Test

Divergence Test

If $\displaystyle\lim_{n \to \infty} a_n \neq 0$, then the series $\displaystyle\sum a_n$ diverges.

Critical Warning — The Converse is False

If $\lim_{n\to\infty} a_n = 0$, the series might still diverge. The harmonic series is the classic example:

$\displaystyle\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$

The terms $\frac{1}{n} \to 0$, but the series diverges. This is one of the most important and counterintuitive facts in calculus. The partial sums grow without bound — incredibly slowly, but without limit.

Other Tests (Preview for Calculus II)

Test	Key Idea	When Useful
Integral Test	If $f(n) = a_n$ and $f$ is continuous, positive, and decreasing, then $\sum a_n$ and $\int_1^\infty f(x)\,dx$ behave the same way.	Series that look like integrals
Comparison Test	If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.	Terms dominated by a known series
Ratio Test	Compute $L = \lim_{n\to\infty} \frac{a_{n+1}}{a_n}$. Converges if $L < 1$, diverges if $L > 1$.	Factorials and exponentials
p-Series	$\displaystyle\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$.	Powers of $n$ in denominator

Section 14.8

Power Series as an Introduction

A power series is an infinite series in which each term contains a power of a variable $x$:

\sum_{n=0}^{\infty} c_n (x - a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \cdots

This is different from everything encountered so far. A power series is not a single number — it is a function of $x$. For each value of $x$, the series either converges (producing a value) or diverges.

Radius of Convergence

Every power series has a radius of convergence $R \geq 0$. The series converges for all $x$ with $|x - a| < R$ and diverges for all $x$ with $|x - a| > R$. The set of $x$-values for which the series converges is called the interval of convergence.

Example 14.8.1

Using the geometric power series

ProblemWrite $f(x) = \dfrac{1}{1+x^2}$ as a power series and identify the interval of convergence.
ThinkRewrite to match $\dfrac{1}{1-r}$: note that $\dfrac{1}{1+x^2} = \dfrac{1}{1 - (-x^2)}$, so $r = -x^2$.
Calculate 1/(1−(−x²)) = Σ (−x²)ⁿ = Σ (−1)ⁿ x²ⁿ = 1 − x² + x⁴ − x⁶ + ···
CheckConverges when $|{-x^2}| < 1$, i.e., $x^2 < 1$, i.e., $|x| < 1$. ✓
InterpretFor $|x| < 1$, $\dfrac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + \cdots$ Integrating term by term gives a series for $\arctan(x)$ — a connection explored fully in Calculus II.

Answer: $\dfrac{1}{1+x^2} = \displaystyle\sum_{n=0}^{\infty} (-1)^n x^{2n}$ for $|x| < 1$

Section 14.9

Functions as Infinite Polynomials

One of the most powerful ideas in mathematics is that functions can be represented as infinite polynomials — at least over an appropriate interval. Polynomials are the simplest functions to work with: they can be evaluated, differentiated, and integrated term by term. The functions $e^x$, $\sin x$, and $\cos x$ have famous power series representations that converge for all real numbers:

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

Graph Instruction

Draw the graph of $y = \sin x$ on $[-2\pi, 2\pi]$. On the same axes, draw $y = x$ and $y = x - \frac{x^3}{6}$. Notice that $y = x$ approximates $\sin x$ near the origin but drifts away quickly. Adding the term $-\frac{x^3}{6}$ significantly improves the approximation. Adding more terms brings the polynomial closer and closer to the true sine curve — illustrating how partial sums converge to the function.

Section 14.10

Taylor Polynomials as Local Approximations

A Taylor polynomial is a polynomial that approximates a function near a specific point. It is constructed using the function's value and its derivatives at that point. If $f$ can be differentiated repeatedly at $x = a$, the $n$-th degree Taylor polynomial centered at $a$ is:

P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n

When $a = 0$, the Taylor polynomial centered at 0 is called a Maclaurin polynomial. The polynomial $P_n(x)$ is constructed so that it matches $f$ and its derivatives of orders 1 through $n$ exactly at $x = a$. No other polynomial of degree $n$ does better at approximating $f$ near $a$.

Term	What it matches
$f(a)$	Function value at $a$
$f'(a)(x-a)$	Slope (first derivative) at $a$
$\frac{f''(a)}{2!}(x-a)^2$	Concavity (second derivative) at $a$
$\frac{f'''(a)}{3!}(x-a)^3$	Rate of change of concavity at $a$

Example 14.10.1

Degree-3 Maclaurin polynomial for $e^x$

ProblemWrite the degree-3 Taylor polynomial for $f(x) = e^x$ centered at $a = 0$.
ThinkAll derivatives of $e^x$ equal $e^x$, so $f(0) = f'(0) = f''(0) = f'''(0) = 1$.
Calculate P₃(x) = f(0) + f'(0)x + f''(0)/2! · x² + f'''(0)/3! · x³ = 1 + x + x²/2 + x³/6
Check$P_3(1) = 1 + 1 + \frac{1}{2} + \frac{1}{6} = \frac{8}{3} \approx 2.667$. True value: $e \approx 2.718$. Close. ✓
InterpretThe polynomial $1 + x + \frac{x^2}{2} + \frac{x^3}{6}$ approximates $e^x$ near $x = 0$. Adding more terms improves accuracy further.

Answer: $P_3(x) = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6}$

Example 14.10.2

Degree-5 Maclaurin polynomial for $\sin x$

ProblemWrite the degree-5 Maclaurin polynomial for $f(x) = \sin x$.
ThinkCompute successive derivatives and evaluate at 0. The pattern of derivatives of $\sin x$ cycles every four steps.
Calculate f(0) = 0, f'(0) = 1, f''(0) = 0, f'''(0) = −1, f⁽⁴⁾(0) = 0, f⁽⁵⁾(0) = 1 P₅(x) = 0 + x + 0 − x³/6 + 0 + x⁵/120 = x − x³/6 + x⁵/120
CheckEvaluate $P_5(0.3)$: $0.3 - \frac{0.027}{6} + \frac{0.00243}{120} = 0.3 - 0.0045 + 0.00002 \approx 0.29552$. True value: $\sin(0.3) \approx 0.29552$. Accurate to 5 decimal places. ✓
InterpretFive terms give high accuracy near the origin. This is precisely how a calculator computes $\sin(0.3)$.

Answer: $P_5(x) = x - \dfrac{x^3}{6} + \dfrac{x^5}{120}$

Example 14.10.3

Degree-3 Taylor polynomial for $\ln x$ centered at $a = 1$

ProblemWrite the degree-3 Taylor polynomial for $f(x) = \ln x$ centered at $a = 1$.
ThinkCannot center at $a = 0$ since $\ln(0)$ is undefined. Center at $a = 1$ instead.
Calculate f(1) = 0, f'(1) = 1, f''(1) = −1, f'''(1) = 2 P₃(x) = 0 + 1·(x−1) + (−1)/2!·(x−1)² + 2/3!·(x−1)³ = (x−1) − (x−1)²/2 + (x−1)³/3
CheckAt $x = 1.2$: $0.2 - 0.02 + 0.00267 \approx 0.18267$. True value: $\ln(1.2) \approx 0.18232$. Close. ✓
InterpretThe center must be chosen where $f$ is defined and differentiable. The polynomial approximates $\ln x$ well near $x = 1$.

Answer: $P_3(x) = (x-1) - \dfrac{(x-1)^2}{2} + \dfrac{(x-1)^3}{3}$

Section 14.11

Error and Approximation

Every Taylor polynomial approximation introduces some remainder (error): the true value of the function minus the polynomial's value, $R_n(x) = f(x) - P_n(x)$. Taylor's Theorem gives the Lagrange form of this remainder:

R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}

where $c$ is some number between $a$ and $x$ (whose exact location we may not know). In practice, we use the error bound:

|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}

where $M$ is an upper bound on $|f^{(n+1)}(t)|$ for all $t$ between $a$ and $x$. This bound tells us the guaranteed worst-case error — exactly what engineers and scientists need.

Example 14.11.1

Estimating error in approximating $e^{0.5}$

ProblemUse $P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$ to approximate $e^{0.5}$ and estimate the error.
ThinkThe fourth derivative of $e^x$ is $e^x$. Its maximum on $[0, 0.5]$ is $e^{0.5} < e < 3$. Use $M = 3$.
Approximate P₃(0.5) = 1 + 0.5 + 0.125 + 0.02083 = 1.64583
Error bound |R₃(0.5)| ≤ 3/4! · (0.5)⁴ = 3/24 · 1/16 = 3/384 ≈ 0.0078
CheckTrue value: $e^{0.5} \approx 1.64872$. Actual error: $0.00289 < 0.0078$. ✓
InterpretThe approximation is guaranteed to be within 0.008 of the true value. The error bound gives a reliable worst-case guarantee, even when the exact error is unknown.

Answer: $e^{0.5} \approx 1.646$ with error $< 0.008$

Section 14.12

Series in Physics, Computing, and Engineering

The ideas of sequences and series appear throughout applied mathematics. They are not abstract curiosities — they are working tools in the sciences.

Computing Transcendental Functions

Computers and calculators cannot evaluate $\sin x$, $e^x$, or $\ln x$ directly from their geometric definitions. Instead, processors use polynomial approximations derived from Taylor series, computed to sufficient precision. The accuracy is controlled by how many terms are included and by the error bound formula.

Fourier Series and Signal Processing

Electrical engineers and audio processors analyze signals using Fourier series: representations of periodic functions as infinite sums of sine and cosine terms. This is the same spirit as Taylor series — expressing a complicated function as an infinite sum of simpler ones — but the building blocks are trigonometric functions rather than polynomials. Fourier series are fundamental to MP3/JPEG compression, MRI imaging, and wireless communications.

Small-Angle Approximation in Physics

A pendulum's equation of motion involves $\sin\theta$. For small angles, physicists replace $\sin\theta$ with the linear Taylor approximation $\theta$, converting a nonlinear differential equation into a solvable linear one. This small-angle approximation appears throughout mechanics, optics, and quantum mechanics. Its accuracy is guaranteed by the error bound.

Financial Mathematics

Geometric series underlie the valuation of bonds, mortgages, annuities, and perpetuities. The present value of a stream of future payments is computed as a geometric series. The pricing formulas used in financial markets are direct applications of the geometric series convergence theorem, as demonstrated in Example 14.6.3.

MGU Library Connection

Convergence tests (ratio, integral, comparison) — developed rigorously in MGU Calculus II
Taylor and Maclaurin series derivations — MGU Calculus II, Power Series chapter
Fourier series — MGU Differential Equations and Scientific Computing pages
Power series methods for differential equations — MGU Differential Equations course
Financial perpetuity and annuity formulas — MGU Economics and Finance Mathematics pages
Series-based numerical methods — MGU Machine Learning Mathematics pages

Section 14.13

Common Series Misunderstandings

Misunderstanding 1 — Terms going to zero guarantees convergence

"If the terms approach zero, the series must converge."

False. The harmonic series $\sum \frac{1}{n}$ has terms approaching 0 but diverges. Terms must shrink fast enough. Zero limit is necessary, not sufficient.

Misunderstanding 2 — Confusing the term sequence with the partial sum sequence

"Since the terms $a_n$ go to zero, the partial sums $S_k$ must also go to zero."

These are two different sequences. $\{a_n\}$ are the terms being added. $\{S_k\}$ is the running total. Convergence of the series depends on $\{S_k\}$, not $\{a_n\}$ alone.

Misunderstanding 3 — A Taylor polynomial equals the function

"The 5th-degree polynomial equals sin x."

A Taylor polynomial is an approximation, not an equality. It equals the function only in the limit as the degree → ∞ (the full Taylor series), and only on the interval of convergence. Always note the polynomial is an approximation.

Misunderstanding 4 — A convergent series works everywhere

"The geometric series for $\frac{1}{1-x}$ works for all $x$."

Power series converge only within their radius of convergence. The series $\sum x^n$ represents $\frac{1}{1-x}$ only for $|x| < 1$. Outside that interval, the series diverges completely.

Misunderstanding 5 — Infinite series can be rearranged freely

"I can rearrange the order of terms in any series."

Finite sums can be rearranged without consequence. For some infinite series (conditionally convergent), changing the order of terms can change the sum — or make it diverge. This is the Riemann Rearrangement Theorem, studied in Calculus II.

Misunderstanding 6 — Convergence is a yes/no question with a universal method

"I just need to learn one test and apply it to every series."

No single test works for all series. Calculus II develops a toolkit — divergence test, integral test, comparison, ratio, alternating series — and students learn which fits each situation.

Misunderstanding 7 — More terms always give better approximation everywhere

"A higher-degree polynomial is more accurate at any x value."

Taylor polynomials improve near the center $a$, but far from the center, even high-degree polynomials can be inaccurate. The error bound includes $|x - a|^{n+1}$, which grows when $x$ is far from $a$. Accuracy is local.

Section 14.14

Readiness for Calculus II and Higher Mathematics

Throughout this textbook, students have developed a complete foundation in single-variable calculus: limits and continuity, derivatives and differentiation rules, applications of derivatives, indefinite and definite integrals, the Fundamental Theorem of Calculus, applications of integration, introductory differential equations, and now sequences and series. These are the tools of classical calculus, developed over three centuries. Students who have genuinely worked through this course have joined a long tradition of mathematical reasoning.

What Calculus II Will Add

Topic Area	What Calculus II Covers
Integration Techniques	Integration by parts, trig integrals, trig substitution, partial fractions, improper integrals
Series Theory (Full)	Divergence, integral, comparison, limit comparison, ratio, root, alternating series tests; conditional vs. absolute convergence
Power Series	Interval of convergence, operations on power series, Taylor and Maclaurin series derivations
Parametric and Polar	New curve representations; area and arc length in these systems

Calculus II Readiness Checklist

✓Compute limits algebraically, graphically, and using L'Hôpital's Rule.
✓Differentiate functions using all standard rules: chain, product, quotient, implicit, transcendental.
✓Set up and solve applied derivative problems: tangent lines, related rates, optimization.
✓Evaluate indefinite integrals using basic rules and substitution.
✓Set up and evaluate definite integrals, including area, volume, and average value applications.
✓Apply the Fundamental Theorem of Calculus, both parts.
✓Solve separable differential equations with initial conditions.
✓Identify the terms of a sequence and determine its limit.
✓Recognize and evaluate a geometric series.
✓Write a Taylor polynomial of a given degree for a given function at a given point.
✓Estimate the error in a Taylor polynomial approximation using the Lagrange remainder.

Chapter 14 — Back Matter

Chapter Summary

Sec 14.1Infinite processes can produce finite results. The half-distance scenario shows that infinitely many terms can sum to a finite value. Whether a process settles down depends on how quickly the terms shrink.
Sec 14.2A sequence is an ordered list $a_1, a_2, a_3, \ldots$ defined by an explicit or recursive formula. Its graph consists of isolated points. Terms may grow, decay, oscillate, or converge.
Sec 14.3A sequence converges to $L$ if $\lim_{n\to\infty} a_n = L$. It diverges otherwise. A sequence that is both monotone and bounded must converge.
Sec 14.4An infinite series is the sum of all terms of a sequence: $\sum_{n=1}^{\infty} a_n$. Its meaning is made precise using partial sums. Sigma notation provides compact representation.
Sec 14.5The $k$-th partial sum is $S_k = a_1 + \cdots + a_k$. The series converges if $\lim_{k\to\infty} S_k = S$ exists and is finite. Telescoping series illustrate this directly.
Sec 14.6A geometric series $\sum ar^n$ converges to $\frac{a}{1-r}$ when $|r| < 1$, and diverges when $|r| \geq 1$. Financial perpetuities are a direct application.
Sec 14.7Key convergence tests are previewed: divergence test, integral test, comparison test, ratio test, p-series result. The harmonic series is the essential warning that terms going to zero does not guarantee convergence.
Sec 14.8A power series $\sum c_n(x-a)^n$ is an infinite polynomial in $x$. It converges on an interval of convergence determined by its radius of convergence $R$. The geometric series is the fundamental example.
Sec 14.9Functions such as $e^x$, $\sin x$, and $\cos x$ equal their power series for all real $x$. These representations allow functions to be computed, differentiated, and integrated term by term.
Sec 14.10The $n$-th degree Taylor polynomial $P_n(x)$ centered at $a$ matches $f$ and its first $n$ derivatives exactly at $a$. Maclaurin polynomials are centered at 0. Each additional term improves local accuracy.
Sec 14.11The Lagrange error bound $|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$ gives a guaranteed worst-case estimate. Error shrinks when $x$ is near $a$ and when more terms are used.
Sec 14.12Series appear in calculator computation of transcendental functions, Fourier analysis in signal processing, small-angle approximation in physics, numerical methods in computing, and financial valuation.
Sec 14.13Seven major misunderstandings: terms to zero does not guarantee convergence; term and partial-sum sequences are different; Taylor polynomials are approximations; power series have limited intervals; rearrangement matters; no universal test exists; accuracy is local.
Sec 14.14Calculus I is a complete single-variable foundation. Calculus II adds integration techniques, full series theory, power series, and parametric/polar coordinates. Every direction in advanced mathematics builds on these tools.

Key Terms Review

sequence

An ordered list of numbers $a_1, a_2, a_3, \ldots$ indexed by a positive integer $n$.

term

An individual number in a sequence. The $n$-th term is $a_n$.

explicit formula

A formula that defines $a_n$ directly as a function of the index $n$.

recursive formula

A formula that defines $a_n$ in terms of one or more previous terms.

convergence

A sequence converges to $L$ if $\lim_{n\to\infty} a_n = L$ for some finite $L$. A series converges if its sequence of partial sums converges.

divergence

A sequence or series diverges if no finite limit exists. Divergence includes growing without bound and oscillating.

bounded sequence

A sequence whose terms all lie between fixed upper and lower bounds $m \leq a_n \leq M$.

monotone sequence

A sequence that is entirely increasing ($a_{n+1} \geq a_n$) or entirely decreasing ($a_{n+1} \leq a_n$).

series

The sum of the terms of a sequence. An infinite series is $\sum_{n=1}^{\infty} a_n$.

partial sum

The finite sum of the first $k$ terms: $S_k = a_1 + a_2 + \cdots + a_k$.

geometric series

A series $\sum ar^n$ with constant ratio $r$. Converges to $\frac{a}{1-r}$ when $|r| < 1$.

common ratio

The constant multiplier $r$ between consecutive terms in a geometric sequence or series.

harmonic series

The series $\sum \frac{1}{n}$. Its terms approach 0, but the series diverges — a foundational example.

power series

An infinite series of the form $\sum c_n(x-a)^n$. It is a function of $x$ that converges on an interval.

radius of convergence

The value $R$ such that the power series converges for $|x - a| < R$ and diverges for $|x - a| > R$.

interval of convergence

The set of all $x$-values for which a power series converges.

Taylor polynomial

The degree-$n$ polynomial matching $f$ and its first $n$ derivatives at a center point $a$.

Maclaurin polynomial

A Taylor polynomial centered at $a = 0$.

remainder

The error $R_n(x) = f(x) - P_n(x)$ between the function and its Taylor polynomial approximation.

Lagrange remainder

The error bound $|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$ for the Taylor polynomial approximation.

p-series

The series $\sum \frac{1}{n^p}$. Converges if $p > 1$, diverges if $p \leq 1$. The harmonic series is $p = 1$.

Fourier series

A representation of a periodic function as an infinite sum of sine and cosine terms. Fundamental to signal processing and engineering.

Sequence Practice

Part AConcept Check

What is the difference between a sequence and a series?
What does it mean for a sequence to converge? Give one example of a convergent sequence and one example of a divergent sequence.
Can a sequence have all positive terms and still diverge? Explain.
What two properties together guarantee that a sequence converges?
Explain why the sequence $a_n = (-1)^n$ diverges, even though its terms are bounded between −1 and 1.

Part BSkill Practice — Sequences

Write the first five terms of each sequence. Determine whether it converges or diverges. If it converges, find the limit.

$a_n = \dfrac{2n}{n+3}$
$a_n = \dfrac{(-1)^n}{n}$
$a_n = \left(\dfrac{3}{4}\right)^n$
$a_n = \sqrt{n}$
$a_n = \dfrac{n^2 - 1}{n^2 + 1}$
$a_n = \dfrac{\ln n}{n}$
$a_n = \cos\!\left(\dfrac{\pi}{n}\right)$
$a_n = \dfrac{5n^3 - 2}{3n^3 + 7}$
$a_n = \dfrac{1}{2^n}$
$a_n = \sin\!\left(\dfrac{1}{n}\right)$
$a_n = n \cdot e^{-n}$
$a_n = \left(1 + \dfrac{1}{n}\right)^n$ (Hint: a famous limit)

Geometric Series Practice

Part CConcept Check

What two pieces of information identify a geometric series?
State the convergence condition for a geometric series and the formula for its sum when it converges.
Give an example of a geometric series that converges and one that diverges, without computing partial sums.

Part DGeometric Series — Evaluate or Classify

Identify the first term and common ratio. Determine whether the series converges and, if so, find the sum.

$\displaystyle\sum_{n=0}^{\infty} \left(\tfrac{1}{3}\right)^n$
$\displaystyle\sum_{n=0}^{\infty} 4 \cdot \left(\tfrac{2}{3}\right)^n$
$\displaystyle\sum_{n=1}^{\infty} \left(\tfrac{5}{4}\right)^n$
$\displaystyle\sum_{n=0}^{\infty} 8 \cdot (-0.5)^n$
$\displaystyle\sum_{n=0}^{\infty} 2 \cdot \left(-\tfrac{3}{2}\right)^n$
$\displaystyle\sum_{n=2}^{\infty} \left(\tfrac{1}{4}\right)^n$

Part EApplied Geometric Series

A ball is dropped from 6 feet and bounces to $\frac{3}{4}$ of its previous height on each bounce. Find the total vertical distance the ball travels (all bounces, both up and down).
A geometric series has first term $a = 5$ and sum $S = 20$. Find the common ratio.
The repeating decimal $0.\overline{36} = 0.363636\ldots$ can be written as an infinite geometric series. Write and evaluate that series to express the decimal as a fraction.

Convergence Concept Questions

Part FReasoning and Explanation

Explain in your own words why the harmonic series $\sum \frac{1}{n}$ diverges even though its terms approach zero.
If $\lim_{n\to\infty} a_n = 0$, can you conclude that $\sum a_n$ converges? Explain with an example supporting your answer.
If $\lim_{n\to\infty} a_n \neq 0$, what can you conclude about $\sum a_n$?
Describe the difference between the sequence of terms $\{a_n\}$ and the sequence of partial sums $\{S_k\}$.
Using the p-series result, determine whether each series converges or diverges: (a) $\sum \frac{1}{n^3}$ (b) $\sum \frac{1}{\sqrt{n}}$ (c) $\sum \frac{1}{n^{0.99}}$ (d) $\sum \frac{1}{n^\pi}$
A student claims: "The series $\sum \frac{1}{n^2}$ must diverge because the harmonic series $\sum \frac{1}{n}$ diverges, and $\frac{1}{n^2} < \frac{1}{n}$." Identify the error.

Approximation Practice

Part GTaylor Polynomial Construction

Degree-4 Maclaurin polynomial for $e^x$
Degree-4 Maclaurin polynomial for $\cos x$
Degree-3 Maclaurin polynomial for $\dfrac{1}{1-x}$
Degree-3 Taylor polynomial for $\sqrt{x}$ at $a = 4$
Degree-2 Taylor polynomial for $\tan x$ at $a = 0$
Degree-3 Maclaurin polynomial for $e^{-x}$

Part HUsing Taylor Polynomials for Approximation

Use the degree-3 Maclaurin polynomial for $\sin x$ to approximate $\sin(0.2)$. Compare with a calculator value.
Use the degree-4 Maclaurin polynomial for $e^x$ to approximate $e^{0.1}$. Compare with a calculator value.
Use the degree-2 Taylor polynomial for $\sqrt{x}$ centered at $a = 4$ to approximate $\sqrt{4.3}$.
Approximate $\cos(0.1)$ using the degree-4 Maclaurin polynomial for $\cos x$.

Part IError Estimation

For the degree-3 Maclaurin polynomial of $\sin x$, estimate the maximum error when approximating $\sin(0.5)$. (Hint: The fourth derivative of $\sin x$ is $\sin x$, so $|f^{(4)}(c)| \leq 1$ for all $c$.)
For the degree-2 Maclaurin polynomial of $e^x$, estimate the maximum error when approximating $e^{0.3}$. (Hint: Bound $e^c$ on $[0, 0.3]$ by $e^{0.3} < 2$.)
How many terms of the Maclaurin series for $\cos x$ are needed to approximate $\cos(1)$ with error less than 0.001? Justify using the error bound.

Applications

Computing $\pi$: Since $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$ and $\arctan(1) = \frac{\pi}{4}$, we can approximate $\pi$ using the first four terms. Compute the approximation. How close is it?
Bouncing Ball: A rubber ball is dropped from 10 meters. Each bounce reaches 60% of the previous height. Write and evaluate a geometric series for the total vertical distance traveled.
Finance: A retirement plan pays $2,000 per month at a monthly interest rate of 0.3%. Use the present value formula for a perpetuity to compute the present value of this income stream.
Population Growth: A bacterial culture grows by 20% each hour starting from 1,000 bacteria. Write the sequence of population values. Is this sequence convergent or divergent? Explain.
Small-Angle Approximation: Using the Taylor polynomial for $\sin\theta$: (a) explain why $\sin\theta \approx \theta$ for small $\theta$; (b) estimate the error at $\theta = 0.1$ radians; (c) find the angle where the error first exceeds 1%.
Taylor Series for $\ln(1+x)$: (a) Compute the degree-5 Maclaurin polynomial for $\ln(1+x)$. (b) Use it to approximate $\ln(1.5)$. (c) Estimate the error. (d) Explain why this series would not work for approximating $\ln(3)$.

Error Analysis

Each problem contains a mathematical error, a conceptual error, or a misinterpretation. Identify the error and write the correct reasoning.

Problem 1

"The series $\sum \frac{1}{n}$ converges because $\lim_{n\to\infty} \frac{1}{n} = 0$."

Identify the logical error and state the correct conclusion.

Problem 2

"The geometric series $\sum (-1.2)^n$ converges because some terms are positive and some are negative, so they cancel."

What is the actual convergence condition? What does sign have to do with it?

Problem 3

Degree-3 Taylor polynomial for $\cos x$: $P_3(x) = 1 - x + x^2 - x^3$

Identify the error. Write the correct degree-3 Taylor polynomial for $\cos x$ centered at 0.

Problem 4

"The polynomial $P_5(x) = x - \frac{x^3}{6} + \frac{x^5}{120}$ equals $\sin x$ for all $x$, because Taylor series represent functions exactly."

What is the correct relationship between a Taylor polynomial and the function it approximates?

Problem 5

Approximating $e^{0.5}$ with $P_2(x)$: "The error is $e - P_2(0.5) = 2.718 - 1.625 = 1.093$."

Identify the conceptual error. What should the student have computed instead?

Chapter 14 Checkpoint

Complete without assistance — assesses all key skills and concepts from Chapter 14

Part A — Sequences (10 points)

(2 pts) Write the first five terms of $a_n = \dfrac{3n-1}{n+2}$.
(2 pts) Find $\lim_{n\to\infty} a_n$ for the sequence in Problem 1. Does it converge or diverge?
(3 pts) Determine whether each sequence converges or diverges. If it converges, find the limit: (a) $a_n = \dfrac{n^3}{e^n}$ (b) $a_n = \cos(n\pi)$ (c) $a_n = \dfrac{4n^2+1}{3n^2-2}$
(3 pts) Explain the difference between a monotone sequence and a bounded sequence. Give an example of a sequence that is bounded but not monotone.

Part B — Geometric Series (20 points)

(4 pts) Determine whether each geometric series converges or diverges. Find the sum if it converges: (a) $\displaystyle\sum_{n=0}^{\infty} \left(\tfrac{2}{5}\right)^n$ (b) $\displaystyle\sum_{n=0}^{\infty} 6 \cdot (-0.9)^n$ (c) $\displaystyle\sum_{n=0}^{\infty} 3 \cdot (1.1)^n$
(4 pts) Write the repeating decimal $0.\overline{36} = 0.363636\ldots$ as an infinite geometric series and find its exact fractional value.
(6 pts) A radioactive substance decays so that 15% of the remaining amount decays each year, starting from 200 grams. (a) Write the sequence formula for the amount after $n$ years. (b) Does this sequence converge? To what value? (c) Set up and evaluate the geometric series for the total amount that decays over infinite time.
(6 pts) Explain in complete sentences why the harmonic series diverges while the geometric series $\sum (\frac{1}{2})^n$ converges. What is the key difference?

Part C — Power Series and Taylor Polynomials (30 points)

(5 pts) Use the geometric series formula to write $f(x) = \dfrac{1}{1-3x}$ as a power series. State the interval of convergence.
(5 pts) Write the degree-4 Maclaurin polynomial for $\cos x$. Evaluate it at $x = 0.5$ and compare with the calculator value of $\cos(0.5)$.
(5 pts) Write the degree-3 Taylor polynomial for $f(x) = \sqrt{1+x}$ centered at $a = 0$.
(5 pts) Write the degree-4 Maclaurin polynomial for $\sin x$. Use it to approximate $\sin\!\left(\frac{\pi}{6}\right)$ and compare with the exact value.
(10 pts) The function $f(x) = e^{-x^2}$ has no elementary antiderivative and is central to probability and statistics. (a) Starting from $e^u = \sum \frac{u^n}{n!}$, substitute $u = -x^2$ to write the first four nonzero terms of the Maclaurin series for $e^{-x^2}$. (b) Use this polynomial to approximate $\displaystyle\int_0^{0.5} e^{-x^2}\,dx$ by integrating term by term. (c) Explain why integrating a Taylor polynomial is useful for functions without elementary antiderivatives.

Part D — Convergence Concepts and Error (20 points)

(5 pts) State the divergence test and give: (a) an example of a series to which it applies, and (b) a series to which it does not apply even though the terms go to zero.
(5 pts) For the degree-3 Maclaurin polynomial of $\sin x$, use the Lagrange error bound to estimate the maximum error in approximating $\sin(0.4)$. Then compute the actual error using a calculator.
(5 pts) A student claims that since $\frac{1}{n^2} < \frac{1}{n}$ and the harmonic series diverges, $\sum \frac{1}{n^2}$ must also diverge. Explain the full error and give the correct conclusion.
(5 pts) Describe in your own words: what a power series is, what the radius of convergence means, and why a series representing a function might only work within a limited interval.

Part E — Bridge to Higher Mathematics (20 points)

(5 pts) Write the first four terms of the power series for $e^x$, $\sin x$, and $\cos x$. Euler's formula states $e^{ix} = \cos x + i\sin x$. Substituting $ix$ for $x$ in the series for $e^x$ and separating real and imaginary parts makes this identity visible. Show the first three terms of both the real and imaginary parts.
(5 pts) Describe three areas of science, engineering, or computing that rely on ideas from this chapter. Be specific about which idea is used in each area.
(10 pts) Reflect: "Calculus is the mathematics of change and accumulation. Sequences and series extend calculus into the study of infinite processes." Write a paragraph explaining how limits (Chapter 2), definite integrals (Chapter 10), and series (Chapter 14) are connected by the same core concept. Use specific mathematical examples.

Answers may be placed in the answer key.

Bridge to Calculus II

The story continues — and deepens

Chapter 14 is the final chapter of this textbook, but it is the opening of a much larger story. In Calculus II, students will develop the complete theory of convergence, master integration techniques, derive the Taylor and Maclaurin series rigorously for all major functions, and apply power series to solve differential equations. The tools introduced briefly here — the ratio test, the integral test, power series, error bounds — will become major topics that students work through at full depth. Students who have understood the ideas in this chapter are prepared for that journey.