Alternating Series Test

The central idea

For a series $\sum(-1)^{n}b_n$ or $\sum(-1)^{n+1}b_n$ with $b_n\ge0$, the alternating-series test guarantees convergence if $b_n$ is eventually nonincreasing and $b_n\to0$. The remainder after $N$ terms satisfies $|R_N|\le b_{N+1}$ and has the sign of the first omitted term.

Definitions, hypotheses, and notation

Monotonicity need only hold after some index because adding or removing finitely many terms changes the sum but not convergence. The error estimate is unusually sharp: it needs no integral and uses the same magnitude sequence already checked. It applies to the alternating sum, not automatically to an absolute series or a series whose signs merely change irregularly.

The theorem is sufficient, not necessary. An alternating series can converge even if its magnitudes are not monotone at every stage. If a hypothesis fails, one must select another test rather than announce divergence, unless the terms themselves fail to approach zero.

Conceptual meaning

Alternating partial sums approach the limit from opposite sides. Decreasing term magnitudes make each correction too small to undo the previous bracket, so the true sum remains trapped between consecutive partial sums.

A dependable method and decision rule

1. Separate the sign pattern from the magnitude $b_n$.
2. Prove $b_n\to0$.
3. Show $b_{n+1}\le b_n$ eventually, by algebra or a derivative of a continuous extension.
4. Conclude convergence, then test absolute convergence if classification is requested.
5. For accuracy, choose $N$ so the first omitted magnitude is at most the tolerance.

Fully worked example

Graphical or geometric meaning

Plot partial sums as points on a number line. Odd and even subsequences form two monotone boundaries whose gap is the next term magnitude. The boundaries close on a common limit as $b_n$ tends to zero.

Common mistakes and why they fail

Verification and reasonableness checks

• Compute a ratio $b_{n+1}/b_n$ or derivative to confirm decrease.
• Verify the requested partial sum convention starts at the stated index.
• Compare consecutive partial sums to see the limiting value bracketed.

Check eventual decrease, then quantify the error

For $\sum(-1)^n b_n$, the test needs $b_n\ge0$, eventual monotonic decrease, and $b_n\to0$. Finitely many irregular early terms do not alter convergence. Once the hypotheses hold, the first omitted magnitude bounds the truncation error, $|R_N|\le b_{N+1}$, and the sum lies between consecutive partial sums. To guarantee tolerance $\varepsilon$, solve $b_{N+1}\le\varepsilon$ and verify the indexing of the first omitted term. Do not apply this error bound to an arbitrary sign-changing series; it relies on alternating, shrinking contributions. Finally, test $\sum|a_n|$ separately if the problem asks whether convergence is absolute or conditional, because the alternating test alone does not answer that stronger question.

Practice

1. Does $\sum(-1)^n/\sqrt n$ converge?
2. Bound the error after 10 terms of $\sum(-1)^{n+1}/n$.
3. If $b_n\not\to0$, can the test conclude convergence?

Connections and next steps

• Absolute and Conditional Convergence
• Divergence Test
• Taylor Series