Infinite Sequences

Infinite Sequences

Part 1 of 7 — Definition & Convergence

A sequence is an ordered list of numbers: $a_1, a_2, a_3, \ldots$

Formally, a sequence is a function $a: \mathbb{N} \to \mathbb{R}$ written as $\{a_n\}$.

Convergence

$\boxed{\lim_{n \to \infty} a_n = L \implies \text{the sequence converges to } L}$

If no finite limit exists, the sequence diverges.

Examples

Key Fact: A sequence converges if and only if its terms approach a single finite number.

Computing Limits of Sequences

Technique 1: Direct substitution (polynomial/rational)

$a_n = \frac{3n^2 + 1}{n^2 - 5} \to \frac{3}{1} = 3$ (divide by highest power of $n$)

Technique 2: Squeeze Theorem

$0 \le \frac{\sin n}{n} \le \frac{1}{n} \to 0$, so $\frac{\sin n}{n} \to 0$.

Technique 3: L'Hôpital's via continuous extension

If $f(x)$ is continuous and $\lim_{x \to \infty} f(x) = L$, then $\lim_{n \to \infty} f(n) = L$.

$a_n = \frac{\ln n}{n}: \lim_{x \to \infty}\frac{\ln x}{x} \overset{\text{L'H}}{=} \lim \frac{1/x}{1} = 0$

Technique 4: Root/ratio for exponential behavior

$a_n = \frac{n!}{n^n} \to 0$ (factorial grows slower than exponential of $n$)

Practice Problems

Concept Checks

Computation

Summary

• Sequence: ordered list $\{a_n\}$
• Converges if $\lim_{n \to \infty} a_n = L$ (finite)
• Techniques: direct comparison, squeeze theorem, L'Hôpital's, growth rate ordering
• Growth rate hierarchy: $\ln n \ll n^p \ll a^n \ll n! \ll n^n$

Next: Part 2 — Monotone sequences and boundedness.

Infinite Sequences — Monotone & Bounded Sequences

Part 2 of 7 — Monotonicity, Bounds, and the Monotone Convergence Theorem

Definitions

Monotone Convergence Theorem

$\boxed{\text{Monotone + Bounded} \implies \text{Convergent}}$

This is one of the most powerful tools for proving convergence without finding the limit.

Key Fact: An increasing sequence that is bounded above must converge. A decreasing sequence that is bounded below must converge.

Testing Monotonicity

Method 1: Difference test $a_{n+1} - a_n > 0 \implies$ increasing; $< 0 \implies$ decreasing.

Method 2: Ratio test (for positive sequences) $a_{n+1}/a_n > 1 \implies$ increasing; $< 1 \implies$ decreasing.

Method 3: Derivative test If $a_n = f(n)$ and $f'(x) > 0$ for $x \ge 1$, then $\{a_n\}$ is increasing.

Example

$a_n = \frac{n}{n+1}$. Is it monotone? Bounded?

$a_{n+1} - a_n = \frac{n+1}{n+2} - \frac{n}{n+1} = \frac{(n+1)^2 - n(n+2)}{(n+2)(n+1)} = \frac{1}{(n+2)(n+1)} > 0$

So $\{a_n\}$ is increasing. Also $a_n < 1$ for all $n$ (bounded above). By MCT, it converges. Indeed, $\lim a_n = 1$.

Practice Problems

Concept Checks

Computation

Summary

• Monotone: always increasing or always decreasing
• Test with: difference, ratio, or derivative
• Monotone Convergence Theorem: Monotone + Bounded $\implies$ Convergent
• This theorem proves existence of a limit without finding it

Next: Part 3 — Recursive sequences and special limits.

Infinite Sequences — Recursive Sequences & Special Limits

Part 3 of 7 — Recursion and Important Limits

Recursive Sequences

A recursive sequence defines $a_{n+1}$ in terms of previous terms:

$a_{n+1} = f(a_n), \quad a_1 = \text{given}$

To find the limit (if it converges), assume $\lim a_n = L$ and solve:

$L = f(L)$

Example

$a_1 = 1$, $a_{n+1} = \sqrt{2 + a_n}$.

If $L$ exists: $L = \sqrt{2 + L} \implies L^2 = 2 + L \implies L^2 - L - 2 = 0 \implies L = 2$ (since $L > 0$).

Must also verify convergence: show the sequence is increasing and bounded above by $2$.

Important Limits to Know

Growth Rate Hierarchy

$\boxed{\ln n \ll n^p \ll a^n \ll n! \ll n^n}$

Each function on the left grows infinitely slower than the one on its right.

Practice Problems

Growth Hierarchy

Recursive Sequence

Summary

• Recursive sequences: find limit by solving $L = f(L)$
• Must separately verify convergence (monotone + bounded)
• Growth hierarchy: $\ln n \ll n^p \ll a^n \ll n! \ll n^n$
• Key limit: $(1 + k/n)^n \to e^k$

Next: Part 4 — Sequences and series connection.

Infinite Sequences — Sequences vs. Series

Part 4 of 7 — The Bridge to Series

Sequence vs. Series

A series is the sum of a sequence. The partial sums form a new sequence:

$S_N = \sum_{n=1}^{N} a_n = a_1 + a_2 + \cdots + a_N$

$\boxed{\sum_{n=1}^\infty a_n = \lim_{N \to \infty} S_N}$

Key Fact: A series converges if and only if the sequence of partial sums converges.

The $n$th Term Test (Divergence Test)

$\boxed{\text{If } \lim_{n \to \infty} a_n \ne 0, \text{ then } \sum a_n \text{ diverges.}}$

Contrapositive: If $\sum a_n$ converges, then $a_n \to 0$.

CAUTION: $a_n \to 0$ does NOT guarantee convergence!

The harmonic series $\sum 1/n$ has $a_n = 1/n \to 0$ but diverges.

Geometric Series

$\sum_{n=0}^\infty r^n = \frac{1}{1-r} \quad \text{if } |r| < 1$

| $|r|$ | Behavior | |-------|----------| | $|r| < 1$ | Converges to $\frac{1}{1-r}$ | | $|r| \ge 1$ | Diverges |

Practice Problems

Key Distinctions

Computation

Summary

• Series = sum of a sequence: $\sum a_n = \lim S_N$
• $n$th Term Test: if $a_n \not\to 0$, series diverges
• $a_n \to 0$ does NOT guarantee convergence
• Geometric series: converges iff $|r| < 1$, sum $= \frac{a}{1-r}$

Next: Part 5 — Telescoping and $p$-series.

Infinite Sequences — Telescoping & $p$-Series

Part 5 of 7 — Special Series Types

Telescoping Series

A telescoping series has partial sums where most terms cancel:

$\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+1}\right) = \lim_{N\to\infty}\left(1 - \frac{1}{N+1}\right) = 1$

How to recognize: Partial fractions often reveal telescoping structure.

$p$-Series

$\boxed{\sum_{n=1}^\infty \frac{1}{n^p} \text{ converges if and only if } p > 1}$

AP Tip: The $p$-series test and geometric series test are the most fundamental — many other tests compare to these.

Telescoping Example

Find $\sum_{n=1}^\infty \frac{1}{n(n+2)}$.

Step 1. Partial fractions: $\frac{1}{n(n+2)} = \frac{1}{2}\left(\frac{1}{n} - \frac{1}{n+2}\right)$

Step 2. Write partial sums: $S_N = \frac{1}{2}\left[\left(1 - \frac{1}{3}\right) + \left(\frac{1}{2} - \frac{1}{4}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \cdots\right]$

Most terms telescope! Surviving terms: $S_N = \frac{1}{2}\left(1 + \frac{1}{2} - \frac{1}{N+1} - \frac{1}{N+2}\right)$

$\sum_{n=1}^\infty \frac{1}{n(n+2)} = \frac{1}{2}\left(\frac{3}{2}\right) = \frac{3}{4}$

Practice Problems

Classification

Computation

Summary

• Telescoping series: write partial sums, identify cancellation
• $p$-series: $\sum 1/n^p$ converges iff $p > 1$
• The harmonic series ($p = 1$) is the critical boundary case
• Partial fractions often reveal hidden telescoping

Next: Part 6 — Problem-Solving Workshop.

Infinite Sequences — Workshop

Part 6 of 7 — Problem-Solving Workshop

Mixed problems covering sequence convergence, series basics, and special types.

Workshop Overview

Problem Set

Problem 1. Does $\sum_{n=1}^\infty \frac{n^2}{n^2 + 1}$ converge?

$a_n = \frac{n^2}{n^2+1} \to 1 \ne 0$. Diverges by the $n$th term test.

Problem 2. Find $\sum_{n=0}^\infty \frac{3}{5^n}$.

Geometric: $a = 3$, $r = 1/5$. Sum $= \frac{3}{1-1/5} = \frac{3}{4/5} = \frac{15}{4}$.

Problem 3. Classify $\sum_{n=1}^\infty \frac{1}{n^\pi}$.

$p$-series with $p = \pi \approx 3.14 > 1$. Converges.

Workshop Questions

Quick Classification

Workshop Computation

Workshop Summary

• $n$th term test: quick divergence check ($a_n \not\to 0$)
• Geometric and $p$-series are the fundamental comparison targets
• Growth hierarchy for sequence limits: $\ln n \ll n^p \ll a^n \ll n!$
• Always check convergence before finding a sum

Next: Part 7 — Comprehensive Review.

Infinite Sequences — Comprehensive Review

Part 7 of 7 — Full Topic Review

Master Reference

AP Tip: The AP BC exam tests sequences primarily through series. Understanding sequence convergence is the foundation for all series work.

Common Pitfalls

1. "$a_n \to 0$ so $\sum a_n$ converges" — FALSE. The harmonic series is the classic counterexample.

2. Confusing the sequence $\{a_n\}$ with the series $\sum a_n$ — one asks about the terms, the other about the sum.

3. Forgetting to verify convergence of recursive sequences — solving $L = f(L)$ only finds CANDIDATES for the limit.

4. Incorrect geometric series formula — remember $\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$ starts at $n = 0$. If starting at $n = 1$: $\frac{ar}{1-r}$.

5. $p$-series boundary — $p = 1$ (harmonic series) DIVERGES. Need $p > 1$ (strictly).

Review Questions

Final Checks

Final Computation

Topic Complete!

You've mastered infinite sequences and the bridge to series:

• Sequence convergence (limits, monotonicity, boundedness)
• Recursive sequences and special limits
• Geometric series, $p$-series, and telescoping series
• The $n$th term test and its limitations

$\boxed{\sum_{n=1}^\infty a_n = \lim_{N \to \infty} S_N \qquad \sum ar^n = \frac{a}{1-r} \;(|r|<1)}$

Up next: Infinite Series — convergence tests (comparison, integral, ratio, root).