Alternating and Absolute Convergence

Cancellation changes the question

Positive-term series accumulate in one direction, so comparison tests can order their totals. Alternating series include positive and negative contributions that may cancel. The series

1-\frac12+\frac13-\frac14+\cdots

does not behave like the positive harmonic series even though its absolute term magnitudes are the same. To reason about signed accumulation, we need to know not only whether terms get small but whether the positive and negative corrections become controlled in a specific way.

Write an alternating series in the form

\sum_{n=1}^{\infty}(-1)^{n-1}b_n,

where $b_n\ge0$ records the magnitude of the $n$ th term. The factor $(-1)^{n-1}$ begins with a positive term; $(-1)^n$ begins with a negative one. The sign convention does not affect convergence, but identifying $b_n$ correctly does. A formula that alternates only after an index shift should be rewritten through its first few terms before applying a test.

L5-V01 — Alternating partial sums bracket a target. Plot partial sums of an alternating decreasing series as dots above and below a dashed horizontal limit line. Show the vertical gap shrinking with later points

The alternating series test

If $b_n$ is eventually decreasing and $b_n\to0$ , then

\sum_{n=1}^{\infty}(-1)^{n-1}b_n

converges. The two conditions work together. Decreasing magnitudes ensure that each correction is no larger than the preceding overshoot, and the zero limit ensures those overshoots become arbitrarily small. The theorem does not require a closed form for the sum.

For the alternating harmonic series, $b_n=1/n$ . The magnitudes decrease and approach zero, so the series converges by the alternating series test. Notice what this does not say: the positive harmonic series still diverges. The alternating signs change the partial-sum process by adding a negative correction after each positive one. Cancellation is a structural feature of the terms, not an informal permission to ignore large contributions.

To show that $b_n$ decreases, compare $b_{n+1}$ and $b_n$ or use a decreasing function after a specified index. A sentence such as “the denominator grows” may be a good observation, but it should lead to an inequality. For $b_n=1/\sqrt{n+2}$ , the denominator increases with $n$ , so $b_{n+1}<b_n$ for all positive indices. Then the limit is zero, completing both requirements.

The next term bounds the alternating error

For an alternating series satisfying the test, the error after $N$ terms is at most the magnitude of the first omitted term:

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

This estimate is unusually direct. The next term is the size of the next overshoot, and all later corrections pull the total back by still smaller amounts. It gives a guaranteed accuracy without computing the unknown infinite sum.

For example, the first omitted magnitude in the alternating harmonic series after $100$ terms is $1/101$ . Therefore $S_{100}$ approximates the sum with error less than $1/101$ , or about $0.00991$ . If an error below $0.001$ is required, choose $N$ so that $1/(N+1)<0.001$ . The answer must state the inequality and solve it; merely saying “take many terms” cannot guarantee a specified tolerance.

L5-V02 — The next term is an error bound. Show a number line with two consecutive partial sums on opposite sides of a marked sum SSS, and bracket their separation by bN+1b_{N+1}bN+1

Absolute convergence is stronger

A series $\sum a_n$ converges absolutely if the positive-term series $\sum|a_n|$ converges. Absolute convergence guarantees ordinary convergence. It means that even after removing all cancellation, the total magnitude remains finite. This stronger behavior makes a series more stable under rearrangement and is often easier to prove with comparison, ratio, or root tests.

The series

\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^2}

converges absolutely because $\sum1/n^2$ converges. By contrast, the alternating harmonic series converges conditionally: it converges with its signs, but $\sum|(-1)^{n-1}/n|=\sum1/n$ diverges. The adjective conditional means the convergence depends on the arrangement of opposing signs; it does not mean the conclusion is uncertain.

L5-V03 — Absolute versus conditional convergence. Draw two compact comparison panels. One shows magnitudes forming a convergent p-series, and the other shows harmonic magnitudes whose signed alternation converges only through cancellation

A dependable decision sequence

First identify whether signs alternate in a regular way. If they do, isolate $b_n$ and test decreasing behavior and the zero limit. Then separately test $\sum|a_n|$ whenever an absolute-convergence classification is requested or when a positive-term method applies easily. Do not call a series conditionally convergent merely because it alternates; it must first be shown to converge while its absolute-value series diverges.

For retrieval, classify $\sum_{n=1}^{\infty}(-1)^{n}/\sqrt{n}$ and find how many terms guarantee an alternating-series approximation error below $0.02$ . Then classify $\sum_{n=1}^{\infty}(-1)^n/n^3$ as absolute or conditional, with a benchmark for the absolute-value series. Explain why the same test cannot be used for a series whose signs do not alternate regularly. The next lesson provides ratio and root tests for series whose dominant structure is exponential or factorial rather than polynomial.