Alternating Series

Alternating Series — The Alternating Series Test

Part 1 of 7 — Foundations

What Is an Alternating Series?

A series whose terms alternate in sign:

$\sum_{n=1}^\infty (-1)^{n+1} b_n = b_1 - b_2 + b_3 - b_4 + \cdots$

or equivalently $\sum (-1)^n b_n$ where $b_n > 0$.

The Alternating Series Test (AST)

$\boxed{\text{If } b_n > 0,\ b_{n+1} \le b_n \text{ (decreasing), and } \lim_{n\to\infty} b_n = 0, \text{ then } \sum (-1)^{n+1} b_n \text{ converges.}}$

The Three Hypotheses

AP Tip: On the AP exam, you must explicitly verify ALL three conditions. Simply stating "by AST" is not sufficient for full credit.

Classic Examples

Alternating Harmonic Series:

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$

Check: $b_n = 1/n > 0$ ✓, $1/(n+1) < 1/n$ ✓, $1/n \to 0$ ✓ → Converges by AST.

Non-Example:

$\sum_{n=1}^\infty \frac{(-1)^{n+1} n}{n+1}$

$b_n = n/(n+1) \to 1 \neq 0$. The third condition FAILS. This series diverges by the Divergence Test.

Why "Eventually Decreasing" Suffices

$b_n$ only needs to be decreasing for $n \ge N$ (some fixed $N$). A finite number of "bad" terms don't affect convergence.

To show decreasing: verify $b_{n+1} < b_n$, or equivalently $b_{n+1}/b_n < 1$, or show $f'(x) < 0$ for the continuous version.

Practice: Applying the AST

Verify AST Conditions

Quick Check

Summary

• Alternating Series Test: three conditions ($b_n > 0$, decreasing, $\to 0$)
• Must verify ALL three explicitly on the AP exam
• "Eventually decreasing" is sufficient
• The AST tells you a series converges but does NOT give the sum

Next: Part 2 — Alternating Series Error Bound (Remainder Estimation).

Alternating Series — Error Bound

Part 2 of 7 — The Alternating Series Remainder

The Error Bound Theorem

If $\sum (-1)^{n+1} b_n$ satisfies the AST conditions and $S$ is the exact sum, then the error after $N$ terms satisfies:

$\boxed{|S - S_N| \le b_{N+1}}$

The error is bounded by the absolute value of the first omitted term.

Why This Works

The partial sums of an alternating series "bracket" the true sum:

$S_1 > S > S_2,\quad S_3 > S > S_4,\quad \ldots$

Each new term overshoots and then corrects, so the error is at most the magnitude of the next term.

Example

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^3}$. Approximate $S$ using 4 terms.

$S_4 = 1 - \frac{1}{8} + \frac{1}{27} - \frac{1}{64} = \frac{1728 - 216 + 64 - 27}{1728} \approx 0.8958$

Error $\le b_5 = 1/125 = 0.008$. So $S \approx 0.896 \pm 0.008$.

AP Tip: This error bound appears almost every year on the BC exam, often in FRQ. Know it cold.

Finding $N$ for a Given Accuracy

Problem: How many terms of $\sum (-1)^{n+1}/n^2$ ensure error $< 0.01$?

Solution: Need $b_{N+1} < 0.01$, i.e., $\frac{1}{(N+1)^2} < 0.01$.

$(N+1)^2 > 100 \implies N+1 > 10 \implies N \ge 10$

So 10 terms give accuracy within $0.01$.

Important Distinctions

Error Bound Practice

Error Analysis

Finding Number of Terms

Summary

• Error $\le |b_{N+1}|$ = first omitted term
• To find $N$ for accuracy $\epsilon$: solve $b_{N+1} < \epsilon$
• Partial sums alternately overestimate and underestimate
• This is one of the MOST tested concepts on the BC exam

Next: Part 3 — Absolute vs. Conditional Convergence in Alternating Series.

Alternating Series — Absolute vs. Conditional Convergence

Part 3 of 7 — Classification of Alternating Series

Review of Definitions

Classification Procedure for Alternating Series

Step 1: Compute $\sum |a_n| = \sum b_n$ (remove the $(-1)^n$ factor).

Step 2: If $\sum b_n$ converges → absolutely convergent (done).

Step 3: If $\sum b_n$ diverges → check AST conditions on original series.

• If AST conditions met → conditionally convergent
• If AST fails → divergent

The Four Essential Examples

| Series | $\sum |a_n|$ | $\sum a_n$ | Classification | |--------|-------------|-----------|---------------| | $\sum (-1)^n/n^2$ | $\sum 1/n^2$ conv. ($p=2$) | Converges | Absolute | | $\sum (-1)^n/n$ | $\sum 1/n$ div. | AST works → conv. | Conditional | | $\sum (-1)^n/\sqrt{n}$ | $\sum 1/\sqrt{n}$ div. ($p=1/2$) | AST works → conv. | Conditional | | $\sum (-1)^n \cdot n/(n+1)$ | Diverges | $a_n \not\to 0$ → div. | Divergent |

Connection to Interval of Convergence

This classification matters most at endpoints of intervals of convergence for power series.

Example: $\sum_{n=1}^\infty \frac{x^n}{n}$

Ratio test: $|x| < 1$ → converges. At the endpoints:

• $x = 1$: $\sum 1/n$ diverges
• $x = -1$: $\sum (-1)^n/n$ converges (conditionally)

So the interval of convergence is $[-1, 1)$.

$\boxed{\text{At endpoints, alternating series often converge conditionally while the positive version diverges.}}$

AP Tip: When finding intervals of convergence, ALWAYS test endpoints separately. Expect at least one endpoint to involve an alternating series.

Classification Practice

Endpoint Classification

Classification Challenge

Summary

• Test $\sum |a_n|$ first; if it converges, you have absolute convergence
• If $\sum |a_n|$ diverges but $\sum a_n$ converges (via AST), it's conditional
• Endpoint testing for power series frequently involves this classification
• $\sum (-1)^n/n^p$: absolute if $p > 1$, conditional if $0 < p \le 1$

Next: Part 4 — Alternating Series and Taylor Polynomials.

Alternating Series — Connections to Taylor Series

Part 4 of 7 — Alternating Series in Taylor/Maclaurin Context

Key Maclaurin Series That Alternate

$\boxed{\text{When a Taylor series alternates, you can use the alternating series error bound for the remainder.}}$

AP Tip: The alternating series error bound is often EASIER to apply than the Lagrange error bound. Use it whenever the series alternates.

Example: Estimating $\cos(0.5)$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

Using 3 terms ($n = 0, 1, 2$):

$P_4(0.5) = 1 - \frac{0.25}{2} + \frac{0.0625}{24} = 1 - 0.125 + 0.002604 = 0.877604$

Error $\le |$first omitted term$| = \frac{(0.5)^6}{720} = \frac{1/64}{720} \approx 0.0000217$

Compare: $\cos(0.5) = 0.877583\ldots$, so actual error $\approx 0.000021$ ✓

Alternating vs. Lagrange Error Bound

For alternating Taylor series, both bounds work:

• Alternating: $|R| \le |\text{next term}|$ — easy!
• Lagrange: $|R_n(x)| \le \frac{M|x-c|^{n+1}}{(n+1)!}$ — need to find $M$

The alternating bound is usually tighter and easier. Use it when available!

Taylor + AST Practice

Error Bound Application

Error Computation

Summary

• Many important Taylor series alternate for certain $x$ values
• When alternating, use AST error bound: easier than Lagrange
• Key functions: $\sin x$, $\cos x$, $e^{-x}$, $\ln(1+x)$, $\arctan x$
• On the AP exam, choose the simpler error bound when both apply

Next: Part 5 — AP Exam Strategies for Alternating Series.

Alternating Series — AP Exam Strategies

Part 5 of 7 — FRQ & MC Techniques

Common AP Question Types

FRQ Template: AST Verification

When the AP exam says "show the series converges using the Alternating Series Test":

1. Identify: "This is an alternating series with $b_n = \ldots$"
2. Positive: "$b_n > 0$ for all $n \ge 1$" ✓
3. Decreasing: "$b_{n+1} \le b_n$ because $f'(x) = \ldots < 0$" ✓
4. Limit: "$\lim_{n\to\infty} b_n = \ldots = 0$" ✓
5. Conclude: "Therefore, $\sum (-1)^{n+1} b_n$ converges by the AST." ✓

AP Tip: Omitting any of the three verifications costs points. Even if one seems "obvious," state it explicitly.

Common AP Mistakes to Avoid

Mistake 1: Forgetting to check $b_n \to 0$

$\sum (-1)^n \frac{n}{n+1}$: Students assume AST applies because it alternates. But $b_n = n/(n+1) \to 1 \neq 0$. Diverges!

Mistake 2: Using $a_n$ instead of $b_n$

$b_n$ is the POSITIVE part. Don't check if $(-1)^n b_n \to 0$ — check if $b_n \to 0$.

Mistake 3: Not showing "decreasing"

Must show $b_{n+1} < b_n$ or use $f'(x) < 0$. Don't just assert it.

Mistake 4: Confusing AST error bound with Lagrange

AP-Style Problems

Exam Strategy Decisions

AP FRQ Practice

Exam Strategy Summary

• Always verify ALL three AST conditions explicitly
• Error bound: $|S - S_N| \le b_{N+1}$
• Overestimate vs. underestimate: depends on parity of $N$ and sign of first term
• Odd $N$ + positive first term → overestimate
• Even $N$ + positive first term → underestimate

Next: Part 6 — Problem-Solving Workshop.

Alternating Series — Problem-Solving Workshop

Part 6 of 7 — Mixed Practice

Work through these problems systematically. For each, identify whether to apply AST, error bound, or classification.

Warm-Up Review

Workshop Problems

Error Bound Workshop

Computation Challenge

Workshop Takeaways

• Check AST conditions systematically
• Error bound problems: solve $b_{N+1} < \epsilon$
• Classification: test $\sum |a_n|$ first
• Factorial denominators converge fast, harmonic-type converge slowly

Next: Part 7 — Comprehensive Review.

Alternating Series — Comprehensive Review

Part 7 of 7 — Full Topic Review

Complete Reference

$\boxed{\text{Alternating series: check conditions, bound the error, classify the convergence.}}$

Comprehensive Review MC

Final Classification Drill

Final Error Bound Problem

Alternating Series — Complete Summary

You've mastered:

• Alternating Series Test — the three conditions and verification
• Error Bound — first omitted term bounds the error
• Over/Underestimate — parity of partial sum count
• Absolute vs. Conditional — classification procedure
• Taylor Series Connection — AST error bound as an alternative to Lagrange
• AP Exam Strategies — full justification requirements

Key Fact: Alternating series and error bounds appear on virtually every BC exam. This is one of the highest-yield topics for your score.

Up Next: Power Series — representation, convergence, and manipulation.