The Central Idea
Consider the sequence
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ \frac15,\ldots \]Its explicit rule is
\[ a_n=\frac1n. \]As \(n\) increases, the terms become smaller and move closer to \(0\).
We express this long-run behavior by writing
\[ \boxed{ \lim_{n\to\infty}a_n=0 }. \]The notation means that the sequence values can be made as close to \(0\) as desired by taking sufficiently large indices.
A Sequence Is a Discrete Function
A sequence assigns one output to each allowed index.
We may write
\[ n\longmapsto a_n. \]If the index set is
\[ \{1,2,3,\ldots\}, \]then the sequence is a function whose domain is the positive integers.
Thus,
\[ \boxed{ \text{sequence} = \text{function on a discrete domain} }. \]Discrete Does Not Mean Random
A discrete function may follow a perfectly regular algebraic rule.
For example,
\[ a_n=3n+2 \]is linear in structure even though it is defined only at integer indices.
Likewise,
\[ b_n=2^n \]is exponential in structure while still being discrete.
Explicit, Recursive, Tabular, and Graphical Views
A sequence may be represented in several equivalent ways.
For example, the sequence
\[ 3,\ 6,\ 12,\ 24,\ldots \]has explicit form
\[ a_n=3\cdot2^{n-1}, \]recursive form
\[ a_1=3,\qquad a_n=2a_{n-1}, \]and ordered pairs
\[ (1,3),\ (2,6),\ (3,12),\ (4,24),\ldots \]Each representation emphasizes different structure.
Recognizing Pattern Families
| Sequence Type | Characteristic Pattern |
|---|---|
| Constant | Terms remain unchanged |
| Arithmetic | Constant first difference |
| Geometric | Constant ratio |
| Quadratic | Constant second difference |
| Fibonacci-type | Later terms depend on multiple earlier terms |
| Iterative | Repeated application of an update rule |
Long-Run Behavior
Once a sequence is defined, we can ask what happens as the index becomes large.
- approaching a finite value;
- growing without bound;
- decreasing without bound;
- oscillating;
- repeating periodically;
- remaining bounded without approaching one value.
Increasing Sequences
A sequence is increasing when later terms are greater than earlier terms.
\[ a_{n+1}>a_n. \]For example,
\[ a_n=n^2 \]gives
\[ 1,\ 4,\ 9,\ 16,\ldots \]and is increasing for positive integer indices.
Decreasing Sequences
A sequence is decreasing when
\[ a_{n+1}gives
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ldots \]and decreases as \(n\) increases.
Monotonic Sequences
A sequence that consistently moves in one direction is called monotonic.
- increasing;
- decreasing;
- nondecreasing;
- nonincreasing.
Monotonicity becomes especially important when studying convergence.
Nondecreasing and Nonincreasing
A sequence is nondecreasing if
\[ a_{n+1}\geq a_n. \]It is nonincreasing if
\[ a_{n+1}\leq a_n. \]Equality is allowed.
Bounded Above
A sequence is bounded above if there is some number \(M\) such that
\[ a_n\leq M \]for every allowed index \(n\).
Bounded Below
A sequence is bounded below if there is some number \(m\) such that
\[ a_n\geq m \]for every allowed index.
Bounded Sequences
A sequence is bounded if it is bounded both above and below.
Equivalently, there is some \(M>0\) such that
\[ \boxed{|a_n|\leq M} \]for every term.
Example: A Bounded Sequence
Consider
\[ a_n=\frac1n. \]For \(n\geq1\),
\[ 0Example: An Unbounded Sequence
Consider
\[ a_n=n^2. \]The values eventually exceed every fixed real number.
Therefore, the sequence is not bounded above.
Oscillating Sequences
A sequence may alternate or move back and forth instead of changing in one direction.
For example,
\[ a_n=(-1)^n \]gives
\[ -1,\ 1,\ -1,\ 1,\ldots \]and oscillates between two values.
Periodic Sequences
A sequence is periodic if its values repeat after a fixed number of indices.
If
\[ a_{n+p}=a_n \]for a positive integer \(p\), then \(p\) is a period.
For \(a_n=(-1)^n\),
\[ a_{n+2}=a_n, \]so the sequence has period \(2\).
Bounded Does Not Mean Convergent
The sequence
\[ (-1)^n \]is bounded because
\[ |(-1)^n|=1. \]But its terms do not approach one single number.
\[ \boxed{ \text{bounded} \not\Rightarrow \text{convergent} }. \]What Is Convergence?
A sequence converges to a number \(L\) if its terms become arbitrarily close to \(L\) as the index becomes sufficiently large.
\[ \boxed{ \lim_{n\to\infty}a_n=L }. \]The number \(L\) is called the limit of the sequence.
Convergence Is About the Tail
A limit describes what happens eventually.
For example,
\[ a_n=\frac{100}{n} \]begins with large values but still satisfies
\[ \boxed{ \lim_{n\to\infty}\frac{100}{n}=0 }. \]Getting Closer Does Not Mean Reaching the Limit
The sequence
\[ a_n=\frac1n \]never equals \(0\) for a positive integer \(n\).
Nevertheless,
\[ \lim_{n\to\infty}\frac1n=0. \]A sequence does not need to reach its limit exactly.
Example: A Convergent Geometric Sequence
Consider
\[ a_n=\left(\frac12\right)^n. \]The terms are
\[ \frac12,\ \frac14,\ \frac18,\ \frac1{16},\ldots \]and become arbitrarily close to \(0\).
\[ \boxed{ \lim_{n\to\infty} \left(\frac12\right)^n =0 }. \]Geometric Sequences and Long-Run Behavior
For
\[ a_n=a_1r^{n-1}, \]if
\[ |r|<1, \]then
\[ r^{n-1}\to0. \]Therefore,
\[ \boxed{a_n\to0}. \]Geometric Growth When \(r>1\)
If
\[ r>1 \]and \(a_1>0\), then
\[ a_n=a_1r^{n-1} \]grows without bound.
It does not converge to a finite real number.
Negative Ratios with Magnitude Less Than One
Consider
\[ a_n=\left(-\frac12\right)^n. \]The signs alternate, but the magnitudes shrink toward \(0\).
\[ \boxed{ \lim_{n\to\infty} \left(-\frac12\right)^n =0 }. \]The Case \(r=-1\)
If
\[ a_n=(-1)^n, \]then the magnitude remains \(1\) and the sign alternates forever.
The sequence is bounded but divergent.
Divergence
A sequence that does not converge to a finite limit is called divergent.
- growth without bound;
- decrease without bound;
- persistent oscillation;
- more complicated nonsettling behavior.
Growth to Positive Infinity
For
\[ a_n=n, \]the values grow beyond every fixed real bound.
\[ \boxed{a_n\to+\infty}. \]This describes unbounded growth, not convergence to an ordinary real number.
Decrease to Negative Infinity
For
\[ a_n=-n^2, \]the terms become arbitrarily negative.
\[ \boxed{a_n\to-\infty}. \]Convergence from a Graph
A discrete graph can suggest convergence when its points move closer to a horizontal level.
For
\[ a_n=\frac1n, \]the points
\[ \left(n,\frac1n\right) \]move closer to the horizontal axis.
Convergence from a Table
Consider
\[ a_n=2+\frac1n. \]| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(2.5\) |
| \(5\) | \(2.2\) |
| \(10\) | \(2.1\) |
| \(100\) | \(2.01\) |
The values suggest
\[ \boxed{a_n\to2}. \]Algebraic Limit Intuition
For
\[ a_n=2+\frac1n, \]since
\[ \frac1n\to0, \]we obtain
\[ \boxed{ \lim_{n\to\infty} \left(2+\frac1n\right) =2 }. \]Fixed Points and Recurrence Limits
Suppose
\[ a_{n+1}=f(a_n). \]If \(a_n\to L\) and \(f\) is continuous at \(L\), then
\[ \boxed{L=f(L)}. \]Thus a limit of such an iteration must be a fixed point of the update rule.
Example: Solving for a Possible Recurrence Limit
Let
\[ a_{n+1}=\frac12a_n+3. \]A possible limit \(L\) satisfies
\[ L=\frac12L+3. \]Therefore,
\[ \boxed{L=6}. \]Finding a Fixed Point Does Not Prove Convergence
Solving
\[ L=f(L) \]identifies a possible limit.
It does not by itself prove that the sequence approaches that value.
Example: A Fixed Point That Is Not Attracting
Consider
\[ a_{n+1}=2a_n. \]The fixed-point equation gives
\[ L=0. \]But with
\[ a_0=1, \]the sequence
\[ 1,\ 2,\ 4,\ 8,\ldots \]moves away from \(0\).
Attracting Behavior
A fixed point is informally called attracting when nearby starting values move toward it under iteration.
For
\[ a_{n+1}=\frac12a_n+3, \]the fixed point \(6\) attracts many starting values.
Monotone and Bounded Behavior
A major theorem for real sequences says that every monotone bounded sequence converges.
\[ \boxed{ \text{monotone} + \text{bounded} \longrightarrow \text{convergent} }. \]Intuitively, if a sequence keeps moving in one direction but cannot move indefinitely, it must approach some limiting level.
Example: Increasing and Bounded
Consider
\[ a_n=1-\frac1n. \]The terms are
\[ 0,\ \frac12,\ \frac23,\ \frac34,\ldots \]They increase and remain below \(1\).
The sequence converges to
\[ \boxed{1}. \]Example: Decreasing and Bounded
The sequence
\[ a_n=2+\frac1n \]decreases and remains above \(2\).
It converges to
\[ \boxed{2}. \]Convergence and Partial Sums
An infinite series
\[ \sum_{k=1}^{\infty}a_k \]is studied through its partial sums
\[ S_n=\sum_{k=1}^{n}a_k. \]The infinite series converges when the sequence
\[ S_1,S_2,S_3,\ldots \]converges to a finite limit.
Example: Geometric Partial Sums
Consider
\[ 1+\frac12+\frac14+\frac18+\cdots. \]The partial sums satisfy
\[ S_n= \frac{1-\left(\frac12\right)^n}{1-\frac12} = 2\left[ 1-\left(\frac12\right)^n \right]. \]Since
\[ \left(\frac12\right)^n\to0, \]we obtain
\[ \boxed{S_n\to2}. \]A First Infinite Geometric-Series Formula
If
\[ |r|<1, \]then
\[ a_1+a_1r+a_1r^2+\cdots \]converges to
\[ \boxed{ \frac{a_1}{1-r} }. \]More detailed study of infinite series belongs to later analysis.
Convergence Is Not the Same as Small Terms
A sequence may have small terms without converging.
For example,
\[ a_n=\frac12(-1)^n \]stays between
\[ -\frac12 \]and
\[ \frac12, \]but oscillates forever.
Eventually Constant Sequences
Some sequences eventually stop changing.
For example,
\[ 3,\ 5,\ 7,\ 7,\ 7,\ 7,\ldots \]converges to
\[ \boxed{7}. \]Finite Changes Do Not Affect the Limit
Convergence depends on the long-run tail of a sequence.
Changing finitely many early terms does not change the limit.
For example,
\[ 1000,\ -50,\ 8,\ \frac14,\ \frac15,\ \frac16,\ldots \]still approaches
\[ \boxed{0}. \]Patterns Can Be Local or Global
A sequence may show one pattern over a short interval and another in the long run.
Long-run analysis therefore requires more than inspecting only the first few terms.
Graphical Pattern Recognition Has Limits
A graph can suggest behavior, but scale matters.
A slowly changing sequence may look flat even when it is still changing. Algebraic analysis should support visual interpretation.
Tables Have Limits Too
A table can display only finitely many values.
Long-run behavior concerns arbitrarily large indices.
Numerical evidence can suggest convergence but does not by itself establish what happens forever.
Common Error: Treating a Large Index as Infinity
The notation
\[ n\to\infty \]does not mean that \(n\) becomes an actual number called infinity.
It means that we examine what happens as \(n\) grows without bound.
Common Error: Assuming Bounded Means Convergent
The sequence
\[ (-1)^n \]is bounded but alternates forever.
Boundedness alone is not sufficient for convergence.
Common Error: Assuming Monotonic Means Convergent
The sequence
\[ a_n=n \]is increasing but unbounded above.
Therefore, it does not converge to a finite real limit.
Common Error: Assuming a Fixed Point Is Automatically the Limit
Solving
\[ L=f(L) \]gives only a candidate fixed point.
The recurrence must still be analyzed to determine whether the sequence approaches it.
Common Error: Thinking a Sequence Must Reach Its Limit
The sequence
\[ \frac1n \]converges to \(0\) without ever equaling \(0\).
Common Error: Confusing Sequence Convergence with Series Convergence
The sequence
\[ a_n=\frac1n \]converges to \(0\).
But the infinite series
\[ \sum_{n=1}^{\infty}\frac1n \]diverges.
A sequence of terms and the series formed from those terms are different mathematical objects.
A Reliable Procedure for Classifying a Sequence Pattern
- Identify the domain and starting index.
- Inspect explicit or recursive structure.
- Check first differences.
- Check ratios where appropriate.
- Build higher finite differences if polynomial structure is suspected.
- Examine signs, repetition, and oscillation.
- Use a table or graph to support the algebraic analysis.
A Reliable Procedure for Introductory Convergence Analysis
- Inspect several terms.
- Check whether the sequence is increasing, decreasing, or oscillating.
- Determine whether it appears bounded.
- Use the explicit rule or recurrence to identify a candidate limit.
- For recurrences, solve the fixed-point equation when appropriate.
- Check whether the sequence actually moves toward that candidate.
- Distinguish finite-limit convergence from unbounded growth.
- Support numerical or graphical evidence with algebra whenever possible.
Example: Full Explicit-Sequence Analysis
Consider
\[ a_n=3+\frac2n. \]Since
\[ \frac2n\to0, \]we obtain
\[ \boxed{ \lim_{n\to\infty} \left(3+\frac2n\right)=3 }. \]Example: Full Recurrence Analysis
Let
\[ a_0=10, \qquad a_{n+1}=\frac12a_n+2. \]The first few terms are
\[ 10,\ 7,\ 5.5,\ 4.75,\ 4.375,\ldots \]A possible limit \(L\) satisfies
\[ L=\frac12L+2. \]Therefore,
\[ \boxed{L=4}. \]The computed terms move toward \(4\), supporting
\[ \boxed{a_n\to4}. \]Practice 1: Identify the Function Type
Explain why
\[ a_n=5n-2,\qquad n=1,2,3,\ldots \]is both a sequence and a discrete function.
Practice 2: Increasing or Decreasing?
Determine whether
\[ a_n=\frac5n \]is increasing or decreasing for \(n\geq1\).
Practice 3: Bounded or Unbounded?
Determine whether
\[ a_n=(-1)^n \]is bounded.
Practice 4: Introductory Limit
Find
\[ \lim_{n\to\infty}\frac3n. \]Practice 5: Shifted Limit
Find
\[ \lim_{n\to\infty} \left( 7+\frac1n \right). \]Practice 6: Geometric Convergence
Determine the limit of
\[ a_n=4\left(\frac13\right)^n. \]Practice 7: Oscillation
Does
\[ a_n=(-1)^n \]converge?
Practice 8: Fixed-Point Candidate
Suppose
\[ a_{n+1}=\frac14a_n+6. \]Find the possible limiting fixed point \(L\).
Practice 9: Monotone but Divergent
Give a reason why
\[ a_n=n^2 \]does not converge to a finite real number.
Practice 10: Identify the Error
A student says:
“The terms of \(1/n\) get very small, so eventually one of them must equal zero.”
Explain the error.
Practice Answers
Practice 1
The rule
\[ a_n=5n-2 \]assigns exactly one output to each allowed integer index \(n\).
Therefore, it is a function on a discrete domain and hence a sequence.
Practice 2
As \(n\) increases, the denominator increases while the numerator remains fixed.
\[ \boxed{ a_n=\frac5n \text{ is decreasing} }. \]Practice 3
Since
\[ |(-1)^n|=1, \]every term lies between \(-1\) and \(1\).
\[ \boxed{\text{bounded}}. \]Practice 4
\[ \boxed{ \lim_{n\to\infty}\frac3n=0 }. \]Practice 5
\[ \boxed{ \lim_{n\to\infty} \left( 7+\frac1n \right) =7 }. \]Practice 6
Since
\[ \left|\frac13\right|<1, \]we have
\[ \boxed{ 4\left(\frac13\right)^n\to0 }. \]Practice 7
No. The sequence alternates between \(-1\) and \(1\), so it never approaches one single value.
\[ \boxed{\text{divergent}}. \]Practice 8
\[ L=\frac14L+6. \] \[ \frac34L=6. \] \[ \boxed{L=8}. \]Practice 9
The sequence \(n^2\) is increasing and unbounded above.
Its values grow beyond every finite real number, so it cannot converge to a finite real limit.
Practice 10
Becoming arbitrarily close to \(0\) does not mean becoming equal to \(0\).
For every positive integer \(n\),
\[ \frac1n>0. \]Thus no term equals \(0\), even though
\[ \boxed{ \frac1n\to0 }. \]What to Remember
A sequence is a function on a discrete index set:
\[ \boxed{n\longmapsto a_n}. \]Convergence to \(L\) is written
\[ \boxed{ \lim_{n\to\infty}a_n=L }. \]- Increasing and decreasing behavior describe direction of change.
- Boundedness describes whether terms remain within fixed limits.
- Boundedness alone does not guarantee convergence.
- Monotonicity alone does not guarantee convergence.
- Monotone and bounded real sequences converge.
- Geometric sequences with \(|r|<1\) converge to \(0\).
- A sequence may approach a limit without ever reaching it.
- Fixed-point equations identify possible recurrence limits but do not by themselves prove convergence.
- Graphs and tables suggest long-run behavior; algebra provides stronger support.
- Infinite series are studied through convergence of their partial-sum sequences.
The main conceptual transition is
\[ \boxed{ \text{individual discrete terms} \longrightarrow \text{long-run behavior} }. \]Where We Go Next
This lesson completes Sequences, Recurrences, and Algebraic Patterns. The section began with the language of terms and indices, developed explicit and recursive definitions, studied arithmetic, geometric, polynomial, and Fibonacci-type patterns, introduced compact sum and product notation, and connected recurrence relations with algorithmic iteration.
The final idea introduced here—convergence—provides a bridge from algebraic patterns into later work on infinite processes, approximation, limits, series, and mathematical analysis.