Alternating Series Test and Error Estimation

Alternating Series Test and Absolute Convergence

By the end of this lesson, you will be able to apply the Alternating Series Test, estimate errors, and distinguish absolute from conditional convergence.

The Alternating Series Test

An alternating series $\sum_{n=1}^\infty (-1)^n b_n$ or $\sum_{n=1}^\infty (-1)^{n+1}b_n$ (with $b_n > 0$) converges if:

1. $b_{n+1} \leq b_n$ for all $n$ (terms are non-increasing)
2. $\lim_{n\to\infty} b_n = 0$

Example: $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ (the alternating harmonic series)

$b_n = 1/n$; $b_{n+1} = 1/(n+1) < 1/n$ ✓; $\lim 1/n = 0$ ✓ → converges.

This series converges to $\ln 2$ (even though $\sum 1/n$ diverges).

Alternating Series Estimation Theorem

If an alternating series satisfies the two conditions above, the error in using the partial sum $S_N$ to approximate the full sum $S$ is bounded by:

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

Example: Estimate $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n!}$ using 4 terms. Error bound:

$b_5 = 1/5! = 1/120 < 0.01$

So using $S_4 = 1 - 1/2 + 1/6 - 1/24 = 5/8 = 0.625$, the error is less than $1/120 \approx 0.0083$.

Absolute vs. Conditional Convergence

Absolute convergence: $\sum |a_n|$ converges → $\sum a_n$ converges (absolutely).

Conditional convergence: $\sum a_n$ converges, but $\sum |a_n|$ diverges.

Theorem: absolute convergence implies convergence. (The converse is false.)

Example:

• $\sum (-1)^n/n^2$: $\sum 1/n^2$ converges (p-series) → absolute convergence.
• $\sum (-1)^n/n$: $\sum 1/n$ diverges; series converges by AST → conditional convergence.

Why does absolute convergence matter? Absolutely convergent series behave like finite sums (rearrangement doesn't change the sum). Conditionally convergent series can be rearranged to converge to any value.

Summary

AST: $(-1)^n b_n$ converges if $b_n$ decreasing and $b_n\to 0$.

$|S - S_N| \leq b_{N+1}$ (next unused term).