The Central Idea
Suppose a sequence has terms
\[ a_1,a_2,a_3,\ldots \]We are often less interested in any one term than in the long-run behavior of the entire process.
If the terms get arbitrarily close to a number \(L\), and remain close once the index is sufficiently large, then we say the sequence converges to \(L\).
\[ \boxed{\lim_{n\to\infty}a_n=L} \]The symbol \(n\to\infty\) does not mean that \(n\) ever becomes infinity. It means that we study what happens as \(n\) takes larger and larger integer values.
What Is a Sequence?
A sequence is a function whose inputs are usually positive integers or nonnegative integers.
\[ a:\mathbb{N}\to\mathbb{R} \]Instead of writing \(a(n)\), sequence notation usually writes
\[ a_n. \]The subscript records the position of the term.
Sequence Versus Set
A sequence is ordered. A set is not.
For example,
\[ 1,\frac12,\frac13,\frac14,\ldots \]is a sequence whose first term is \(1\), second term is \(1/2\), and so on. Reordering the terms would create a different sequence even if the same values appeared.
Explicit Sequence Formulas
An explicit formula gives the \(n\)-th term directly.
For example,
\[ a_n=\frac{1}{n} \]produces
\[ 1,\frac12,\frac13,\frac14,\ldots \]Because \(1/n\) becomes smaller as \(n\) grows, this sequence approaches zero.
\[ \boxed{\lim_{n\to\infty}\frac1n=0} \]Recursive Sequences
A recursive definition generates each new term from earlier terms.
For example,
\[ a_1=8, \qquad a_{n+1}=\frac{a_n+2}{2}. \]The first few terms are
\[ 8,\ 5,\ \frac72,\ \frac{11}{4},\ \ldots \]The values move toward \(2\). Recursive sequences are especially important because many numerical algorithms and dynamical models are naturally iterative.
Reading a Sequence from a Table
A table can make long-run behavior visible before any formal proof is attempted.
| \(n\) | \(a_n=1/n\) |
|---|---|
| 1 | 1 |
| 10 | 0.1 |
| 100 | 0.01 |
| 1000 | 0.001 |
| 10000 | 0.0001 |
The table suggests that the terms can be made as close to zero as desired by choosing a sufficiently large index.
Graphing a Sequence
A real-valued sequence can be graphed as the discrete points
\[ (n,a_n). \]The graph is not normally connected by a continuous curve because the sequence is defined only at integer indices. A convergent sequence appears to have points that crowd closer and closer to a horizontal level \(y=L\).
What Convergence Really Requires
It is not enough for a sequence to get close to \(L\) once. It must eventually remain close.
For convergence to \(L\), every tolerance band around \(L\), no matter how narrow, must eventually contain all later terms.
This word eventually is one of the most important ideas in analysis.
The Formal Definition of Sequence Convergence
The rigorous definition uses a positive tolerance \(\varepsilon\).
We say
\[ \lim_{n\to\infty}a_n=L \]if for every \(\varepsilon>0\), there exists an integer \(N\) such that whenever \(n\ge N\),
\[ |a_n-L|<\varepsilon. \]In plain language: choose any acceptable error. We can go far enough along the sequence that every later term stays within that error of the limit.
Understanding \(\varepsilon\) and \(N\)
The number \(\varepsilon\) describes how close we demand the terms to be to the proposed limit.
The index \(N\) describes how far along the sequence we must go before that level of accuracy is guaranteed.
A smaller \(\varepsilon\) may require a larger \(N\).
Example: Proving \(1/n\to 0\)
Let
\[ a_n=\frac1n. \]We want
\[ \left|\frac1n-0\right|<\varepsilon. \]Since \(n>1/\varepsilon\) implies \(1/n<\varepsilon\), it is enough to choose an integer \(N\) satisfying
\[ N>\frac1\varepsilon. \]Then every \(n\ge N\) satisfies the required inequality. Therefore,
\[ \boxed{\lim_{n\to\infty}\frac1n=0}. \]Convergent and Divergent Sequences
A sequence is convergent if it approaches a finite real number.
If no finite limit exists, the sequence is divergent.
Divergence can happen in several ways. A sequence may grow without bound, decrease without bound, oscillate, or behave irregularly.
Divergence by Unbounded Growth
Consider
\[ a_n=n^2. \]The terms grow larger without bound:
\[ 1,4,9,16,25,\ldots \]We often write
\[ a_n\to\infty, \]but \(+\infty\) is not a finite real-number limit. In the ordinary real sense, the sequence diverges.
Divergence by Oscillation
Consider
\[ a_n=(-1)^n. \]The sequence alternates between
\[ -1,1,-1,1,\ldots \]It never settles near one finite number, so
\[ \boxed{(-1)^n\text{ diverges}.} \]Oscillation Can Still Converge
Alternation alone does not force divergence.
Consider
\[ a_n=\frac{(-1)^n}{n}. \]The signs alternate, but the magnitudes shrink toward zero:
\[ |a_n|=\frac1n\to0. \]Therefore,
\[ \boxed{\lim_{n\to\infty}\frac{(-1)^n}{n}=0}. \]A Limit, If It Exists, Is Unique
A sequence cannot converge to two different real numbers.
If
\[ a_n\to L \qquad\text{and}\qquad a_n\to M, \]then necessarily
\[ \boxed{L=M}. \]This uniqueness makes the phrase the limit meaningful.
Finite Changes Do Not Affect the Limit
Convergence depends on the eventual behavior of the sequence.
Changing the first term, the first ten terms, or any other finite number of terms does not change a sequence limit.
This is useful because infinite-process questions are fundamentally about the tail of the sequence, not its opening entries.
Bounded Sequences
A sequence is bounded if all of its terms stay inside some fixed finite interval.
That is, there exists \(M>0\) such that
\[ |a_n|\le M \]for every index \(n\).
Every Convergent Sequence Is Bounded
If \(a_n\to L\), then the terms eventually remain near \(L\). The finitely many earlier terms are also finite, so one sufficiently large bound can contain the entire sequence.
Therefore,
\[ \boxed{\text{convergent}\Longrightarrow\text{bounded}.} \]The converse is false.
Bounded Does Not Mean Convergent
The sequence
\[ (-1)^n \]is bounded because every term lies between \(-1\) and \(1\), but it does not converge.
Boundedness controls size. Convergence controls long-run location.
Monotone Sequences
A sequence is increasing if
\[ a_{n+1}\ge a_n \]and decreasing if
\[ a_{n+1}\le a_n. \]A sequence that is always increasing or always decreasing is called monotone.
The Monotone Convergence Principle
One of the most useful convergence tools is the monotone convergence theorem for sequences:
If a sequence is monotone and bounded, then it converges.
More specifically, an increasing sequence that is bounded above converges, and a decreasing sequence that is bounded below converges.
Example: A Bounded Increasing Sequence
Consider
\[ a_n=1-\frac1n. \]The terms are
\[ 0,\frac12,\frac23,\frac34,\ldots \]They increase and never exceed \(1\). In fact,
\[ \lim_{n\to\infty}\left(1-\frac1n\right)=1. \]Algebraic Limit Laws for Sequences
Suppose
\[ a_n\to L \qquad\text{and}\qquad b_n\to M. \]Then, when the expressions are defined,
\[ a_n+b_n\to L+M, \] \[ a_n-b_n\to L-M, \] \[ a_nb_n\to LM, \]and if \(M\ne0\),
\[ \frac{a_n}{b_n}\to\frac{L}{M}. \]Constant multiples and integer powers behave as expected as well.
Example: Using Limit Laws
Evaluate
\[ \lim_{n\to\infty}\frac{3n+5}{n}. \]Divide by \(n\):
\[ \frac{3n+5}{n}=3+\frac5n. \]Since \(5/n\to0\),
\[ \boxed{\lim_{n\to\infty}\frac{3n+5}{n}=3}. \]Rational Sequences and Dominant Powers
For a rational expression in \(n\), compare the highest powers in the numerator and denominator.
For example,
\[ \frac{4n^2-3n+1}{2n^2+7} \]can be divided by \(n^2\):
\[ \frac{4-3/n+1/n^2}{2+7/n^2}. \]The vanishing terms approach zero, so
\[ \boxed{\lim_{n\to\infty}\frac{4n^2-3n+1}{2n^2+7}=2}. \]Three Common Degree Patterns
For rational sequences of polynomials in \(n\):
- If the numerator degree is smaller, the limit is usually \(0\).
- If the degrees match, the limit is the ratio of leading coefficients.
- If the numerator degree is larger, the sequence is typically unbounded in magnitude.
Sign and parity still matter in the unbounded case.
Geometric Sequences
A geometric sequence has the form
\[ a_n=Cr^n. \]Its long-run behavior depends strongly on the magnitude of \(r\).
- If \(|r|<1\), then \(r^n\to0\).
- If \(r=1\), then \(r^n=1\).
- If \(r=-1\), the terms oscillate.
- If \(|r|>1\), the magnitude grows without bound.
Example: Exponential Decay in a Sequence
Consider
\[ a_n=7\left(\frac23\right)^n. \]Because
\[ \left|\frac23\right|<1, \]repeated multiplication shrinks the magnitude toward zero. Therefore,
\[ \boxed{a_n\to0}. \]The Squeeze Principle for Sequences
Suppose that for all sufficiently large \(n\),
\[ b_n\le a_n\le c_n, \]and
\[ b_n\to L, \qquad c_n\to L. \]Then
\[ \boxed{a_n\to L}. \]Example: Squeezing an Oscillating Sequence
Consider
\[ a_n=\frac{\sin n}{n}. \]Since
\[ -1\le\sin n\le1, \]dividing by positive \(n\) gives
\[ -\frac1n\le\frac{\sin n}{n}\le\frac1n. \]Both outer sequences approach zero, so
\[ \boxed{\lim_{n\to\infty}\frac{\sin n}{n}=0}. \]Absolute-Value Comparison
A useful special case of squeezing is:
if
\[ |a_n|\le b_n \]eventually, and \(b_n\to0\), then
\[ \boxed{a_n\to0}. \]This is often the fastest way to handle oscillating factors whose magnitude is controlled by a shrinking sequence.
Sequences Defined by Continuous Functions
If \(a_n\to L\) and a function \(f\) is continuous at \(L\), then
\[ f(a_n)\to f(L). \]For example, since
\[ 1+\frac1n\to1, \]continuity of the square root gives
\[ \sqrt{1+\frac1n}\to1. \]Sequences and Function Limits
A sequence is discrete, but many sequence limits can be recognized by comparing \(a_n=f(n)\) with the continuous function \(f(x)\).
If
\[ \lim_{x\to\infty}f(x)=L, \]then the sampled sequence \(a_n=f(n)\) also approaches \(L\).
This creates one of the central bridges from discrete processes to continuous analysis.
Subsequences
A subsequence is formed by selecting terms from a sequence while preserving their original order.
If \((a_n)\) is a sequence and
\[ n_1<n_2<n_3<\cdots, \]then
\[ a_{n_1},a_{n_2},a_{n_3},\ldots \]is a subsequence.
Convergence Passes to Subsequences
If a sequence converges to \(L\), then every subsequence also converges to \(L\).
\[ a_n\to L \quad\Longrightarrow\quad a_{n_k}\to L. \]This gives a powerful way to prove divergence: find two subsequences that approach different limits.
Using Subsequences to Prove Divergence
For
\[ a_n=(-1)^n, \]the even-index subsequence satisfies
\[ a_{2k}=1, \]while the odd-index subsequence satisfies
\[ a_{2k-1}=-1. \]Since these subsequences have different limits, the original sequence cannot converge.
Limit Inferior and Limit Superior: A Preview
Some bounded sequences do not converge but still have meaningful long-run lower and upper accumulation levels. Advanced analysis records these using the limit inferior and limit superior.
For \((-1)^n\), those levels are \(-1\) and \(1\). We will not develop the full theory here, but the idea helps organize oscillatory behavior that has no single limit.
Cauchy Sequences: Convergence Without Naming the Limit First
A sequence is called Cauchy if its terms eventually become arbitrarily close to one another.
Formally, for every \(\varepsilon>0\), there exists \(N\) such that whenever \(m,n\ge N\),
\[ |a_m-a_n|<\varepsilon. \]In the real numbers, a sequence converges if and only if it is Cauchy. This deeper criterion becomes important when the limit is not known in advance.
Recursive Limits: A Common Strategy
Suppose a recursive sequence is known to converge and satisfies
\[ a_{n+1}=F(a_n). \]If \(F\) is continuous and \(a_n\to L\), then passing to the limit gives
\[ L=F(L). \]The limit must therefore be a fixed point of the recurrence.
Example: Solving for a Recursive Limit
Suppose
\[ a_{n+1}=\frac{a_n+6}{2} \]and suppose the sequence has been shown to converge to \(L\). Then
\[ L=\frac{L+6}{2}. \]Solving,
\[ 2L=L+6 \]gives
\[ \boxed{L=6}. \]Solving the fixed-point equation identifies a possible limit. It does not, by itself, prove that the sequence actually converges.
Why Finding a Candidate Limit Is Not Enough
A recurrence may have a fixed point without every sequence generated by the recurrence approaching that point.
A complete argument often requires additional control such as boundedness, monotonicity, contraction behavior, or another convergence theorem.
Numerical Exploration
Calculating many terms is useful for forming a conjecture about convergence.
For example, a spreadsheet, calculator, or short program can generate
\[ a_1,a_2,\ldots,a_{1000} \]and reveal whether the values appear to stabilize.
Numerical evidence is diagnostic, not conclusive. Very slow divergence or late oscillation can be missed by finite computation.
Slow Convergence
Not all convergent sequences approach their limits at the same rate.
The sequence
\[ \frac1n \]approaches zero much more slowly than
\[ \left(\frac12\right)^n. \]Rate of convergence becomes important in numerical analysis because it determines how many steps are needed to achieve a desired accuracy.
Error as Distance from the Limit
If \(a_n\to L\), define the error at step \(n\) by
\[ E_n=|a_n-L|. \]Convergence is exactly the statement
\[ E_n\to0. \]Thinking in terms of error connects abstract convergence directly to approximation algorithms.
Rate Comparison: Geometric Versus Algebraic Decay
For many fixed \(0<r<1\), geometric decay
\[ r^n \]eventually beats algebraic decay such as
\[ \frac1{n^p},\qquad p>0. \]Both go to zero, but geometric sequences generally reach small tolerances much faster.
Important Benchmark Limits
The following sequence limits appear repeatedly:
\[ \frac1{n^p}\to0\qquad(p>0), \] \[ r^n\to0\qquad(|r|<1), \] \[ n^{1/n}\to1, \] \[ \left(1+\frac1n\right)^n\to e. \]The final two require more analysis than the first two, but they are worth recognizing early because they recur throughout calculus and analysis.
Common Error: Looking Only at Early Terms
The first few terms can be misleading. Convergence concerns what happens eventually, not what happens initially.
A sequence may rise for a long time and later fall, or appear stable for many terms and then drift.
Use early terms to explore behavior, but use mathematical structure to establish the conclusion.
Common Error: Treating Infinity as an Index
Expressions such as
\[ a_\infty \]are generally not ordinary sequence terms. The index \(n\) remains finite. The notation \(n\to\infty\) describes a limiting process.
Common Error: Substituting \(\infty\)
To analyze
\[ \frac{3n+1}{n+4}, \]we do not substitute \(n=\infty\) as though infinity were a real number. Instead, divide numerator and denominator by \(n\):
\[ \frac{3+1/n}{1+4/n}\to\frac31=3. \]Common Error: Bounded Implies Convergent
Boundedness is necessary for convergence but not sufficient.
The sequence \((-1)^n\) stays inside \([-1,1]\) and still diverges.
Boundedness becomes sufficient when combined with additional structure, such as monotonicity.
Common Error: Finding a Recursive Fixed Point and Stopping
If a recurrence produces the equation
\[ L=F(L), \]solving this equation identifies a candidate limit. It does not prove convergence.
First establish that the sequence converges, or use a theorem whose hypotheses guarantee convergence.
A Reliable Procedure for Evaluating Sequence Limits
- Write the general term clearly.
- Identify the dominant structure: rational, geometric, oscillatory, recursive, or other.
- Check whether standard benchmark limits apply.
- Use algebraic limit laws when the component limits exist.
- For rational expressions, divide by the dominant power of \(n\).
- For oscillating bounded factors, look for an absolute-value comparison or squeeze argument.
- For recursive sequences, investigate boundedness and monotonicity before solving a fixed-point equation.
- Use subsequences when divergence by oscillation is suspected.
- Use an \(\varepsilon\)-\(N\) argument when a formal proof is required.
Example: Full Rational Workflow
Evaluate
\[ \lim_{n\to\infty}\frac{5n^3-2n+7}{2n^3+n^2-1}. \]The highest power is \(n^3\). Divide numerator and denominator by \(n^3\):
\[ \frac{5-2/n^2+7/n^3}{2+1/n-1/n^3}. \]Every reciprocal power of \(n\) approaches zero, so
\[ \boxed{\frac{5}{2}}. \]Example: Full Squeeze Workflow
Evaluate
\[ a_n=\frac{\cos(n^2)}{\sqrt n}. \]Because
\[ |\cos(n^2)|\le1, \]we have
\[ \left|\frac{\cos(n^2)}{\sqrt n}\right| \le \frac1{\sqrt n}. \]Since \(1/\sqrt n\to0\),
\[ \boxed{\frac{\cos(n^2)}{\sqrt n}\to0}. \]Example: A Monotone Recursive Sequence
Let
\[ a_1=1, \qquad a_{n+1}=\frac{a_n+3}{2}. \]The first terms are
\[ 1,2,\frac52,\frac{11}{4},\ldots \]They suggest that the sequence increases toward \(3\). A full proof can show by induction that \(a_n<3\) and that \(a_{n+1}>a_n\). Thus the sequence is increasing and bounded above, so it converges.
Let the limit be \(L\). Then
\[ L=\frac{L+3}{2}, \]so
\[ \boxed{L=3}. \]Practice 1: Reciprocal Power
Determine whether the sequence converges and, if so, find its limit:
\[ a_n=\frac{7}{n^3}. \]Practice 2: Rational Sequence
Evaluate
\[ \lim_{n\to\infty}\frac{6n^2+5n-1}{3n^2+8}. \]Practice 3: Lower-Degree Numerator
Evaluate
\[ \lim_{n\to\infty}\frac{4n+9}{n^2+1}. \]Practice 4: Geometric Sequence
Determine the limit:
\[ a_n=5\left(-\frac14\right)^n. \]Practice 5: Oscillation
Determine whether the sequence converges:
\[ a_n=3(-1)^n. \]Practice 6: Oscillation with Shrinking Magnitude
Evaluate
\[ \lim_{n\to\infty}\frac{(-1)^n}{n^2}. \]Practice 7: Squeeze Principle
Evaluate
\[ \lim_{n\to\infty}\frac{\sin(3n)}{n}. \]Practice 8: Boundedness and Convergence
Is the statement true or false?
Every bounded sequence converges.
Give a reason or counterexample.
Practice 9: Recursive Candidate Limit
Suppose a sequence converges and satisfies
\[ a_{n+1}=\frac{2a_n+5}{3}. \]Find its possible limit.
Practice 10: Formal Tolerance
For
\[ a_n=\frac1n, \]find an integer \(N\) that guarantees
\[ |a_n|<0.001 \]whenever \(n\ge N\).
Practice Answers
Practice 1
Since \(1/n^3\to0\),
\[ \boxed{\frac7{n^3}\to0}. \]Practice 2
The numerator and denominator have the same degree, so the limit is the ratio of leading coefficients:
\[ \boxed{2}. \]Practice 3
Divide numerator and denominator by \(n^2\):
\[ \frac{4/n+9/n^2}{1+1/n^2}\to0. \]Therefore,
\[ \boxed{0}. \]Practice 4
Since \(|-1/4|<1\),
\[ \boxed{5\left(-\frac14\right)^n\to0}. \]Practice 5
The sequence alternates between \(-3\) and \(3\). It does not approach one number, so it diverges.
Practice 6
Since
\[ \left|\frac{(-1)^n}{n^2}\right|=\frac1{n^2}\to0, \]we have
\[ \boxed{\frac{(-1)^n}{n^2}\to0}. \]Practice 7
Because \(|\sin(3n)|\le1\),
\[ \left|\frac{\sin(3n)}n\right|\le\frac1n\to0. \]Therefore,
\[ \boxed{0}. \]Practice 8
The statement is false. The sequence
\[ a_n=(-1)^n \]is bounded but divergent.
Practice 9
If \(a_n\to L\), then
\[ L=\frac{2L+5}{3}. \]Thus
\[ 3L=2L+5, \]so the possible limit is
\[ \boxed{L=5}. \]Practice 10
We need
\[ \frac1n<0.001=\frac1{1000}. \]Any \(n>1000\) works. One convenient choice is
\[ \boxed{N=1001}. \]What to Remember
A sequence converges to \(L\) when its terms eventually remain arbitrarily close to \(L\):
\[ \boxed{\lim_{n\to\infty}a_n=L}. \]- Convergence is about eventual behavior, not early terms.
- A convergent sequence has one unique limit.
- Every convergent sequence is bounded, but not every bounded sequence converges.
- Monotone and bounded sequences converge.
- Limit laws simplify many algebraic sequence limits.
- The squeeze principle handles many bounded oscillations.
- Different subsequential limits prove divergence.
- For recursive sequences, a fixed-point equation finds a candidate limit only after convergence is justified.
- The formal \(\varepsilon\)-\(N\) definition makes the phrase “arbitrarily close” precise.
Where We Go Next
Sequence convergence studies the long-run behavior of individual terms. The next step is to accumulate terms through repeated addition.
The next lesson, Infinite Series, asks whether partial sums such as
\[ a_1+a_2+\cdots+a_n \]approach a finite limit as more and more terms are included. That shift from terms to partial sums begins the systematic study of infinite series.