A conditionally convergent series is a warning sign. Its terms go to zero, its partial sums settle to a finite value, and yet the positive mass and the negative mass are both infinite when separated. The convergence is produced by cancellation rather than by small total size. Alternating series are the first systematic setting where this cancellation can be controlled.

The simplest mystery is the harmonic series. The series $\sum_{n=1}^{\infty} 1/n$ diverges, so making the terms smaller only as $1/n$ is not enough for convergence. But inserting alternating signs changes the behaviour completely:

\begin{align*} 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. \end{align*}

The terms still have the same absolute values as the harmonic series. The difference is that each positive term is followed by a slightly smaller negative correction, and each negative term is followed by a still smaller positive correction. The partial sums are trapped from above and below.

[example: The Alternating Harmonic Series] Consider the partial sums \begin{align*} s_N=\sum_{n=1}^{N}\frac{(-1)^{n+1}}{n}. \end{align*} For $m\in\mathbb{N}$, the next two terms after $s_{2m}$ are positive and then negative, so \begin{align*} s_{2m+2}=s_{2m}+\frac{1}{2m+1}-\frac{1}{2m+2}. \end{align*} Hence \begin{align*} s_{2m+2}-s_{2m}=\frac{1}{2m+1}-\frac{1}{2m+2}=\frac{1}{(2m+1)(2m+2)}>0. \end{align*} Thus the even partial sums $(s_{2m})$ are increasing. Similarly, the next two terms after $s_{2m+1}$ are negative and then positive, so \begin{align*} s_{2m+3}=s_{2m+1}-\frac{1}{2m+2}+\frac{1}{2m+3}. \end{align*} Therefore \begin{align*} s_{2m+3}-s_{2m+1}=-\frac{1}{2m+2}+\frac{1}{2m+3}=-\frac{1}{(2m+2)(2m+3)}<0. \end{align*} Thus the odd partial sums $(s_{2m-1})$ are decreasing. The two subsequences are interlaced. Since \begin{align*} s_{2m+1}=s_{2m}+\frac{1}{2m+1}, \end{align*} we have \begin{align*} s_{2m}\le s_{2m+1}. \end{align*} Also \begin{align*} s_{2m-1}=s_{2m}+\frac{1}{2m}, \end{align*} so every even partial sum lies below the preceding odd partial sum. Hence the increasing sequence $(s_{2m})$ is bounded above, for instance by $s_1$, and the decreasing sequence $(s_{2m-1})$ is bounded below, for instance by $s_2$. By the monotone convergence principle, both subsequences have limits. Finally, their gap is exactly the last added positive term: \begin{align*} s_{2m+1}-s_{2m}=\frac{1}{2m+1}. \end{align*} Since $\frac{1}{2m+1}\to 0$, the two subsequential limits are equal. Therefore the alternating harmonic series converges, even though the associated positive harmonic series $\sum_{n=1}^{\infty}1/n$ diverges; the convergence comes from the ordered cancellation between successive positive and negative terms. [/example]

This page studies alternating series as a child concept of [series](/page/Series). The parent theory asks when the partial sums of $\sum a_n$ converge. Here the special structure is that the signs alternate and the magnitudes usually decrease. The point is not only to prove convergence; it is to obtain a usable error estimate and to understand why conditional convergence is delicate.

## Definition

A general series allows arbitrary signs, so there is no reason for cancellation to occur in an orderly way. Alternating series isolate the most regular possible sign pattern: plus, minus, plus, minus, or the same pattern shifted by one place. The remaining data are the non-negative magnitudes of the terms.

[definition: Alternating Series] An alternating series is a real series of the form \begin{align*} \sum_{n=1}^{\infty} (-1)^{n+1} b_n \end{align*} or \begin{align*} \sum_{n=1}^{\infty} (-1)^n b_n, \end{align*} where $b_n \ge 0$ for every $n \in \mathbb{N}$. [/definition]

The sign pattern still leaves a major question unanswered: when is the cancellation orderly enough to force convergence? If later corrections can be larger than earlier ones, the partial sums need not settle. The useful special case is therefore the one where the magnitudes shrink monotonically to zero.

[definition: Decreasing Alternating Series] A decreasing alternating series is an alternating series \begin{align*} \sum_{n=1}^{\infty} (-1)^{n+1} b_n \end{align*} with $b_n \ge 0$ for every $n \in \mathbb{N}$, $b_{n+1} \le b_n$ for every $n \in \mathbb{N}$, and $b_n \to 0$ as $n \to \infty$. [/definition]

The word decreasing is often used in the non-strict sense: equality between consecutive terms is allowed. What matters is that the later corrections never exceed the earlier errors they are meant to correct.

To compare an infinite alternating sum with its proposed value, we need to watch the finite approximations separately according to parity. The odd approximations and even approximations usually lie on opposite sides of the sum, so naming them lets us state the bracketing mechanism precisely.

[definition: Alternating Partial Sums] For an alternating series $\sum_{n=1}^{\infty} (-1)^{n+1}b_n$, the $N$-th partial sum is \begin{align*} s_N = \sum_{n=1}^{N} (-1)^{n+1}b_n. \end{align*} The even partial sums are $(s_{2m})_{m=1}^{\infty}$, and the odd partial sums are $(s_{2m-1})_{m=1}^{\infty}$. [/definition]

The whole theory rests on the geometry of these two subsequences on the real line. Even partial sums approach the eventual answer from one side, odd partial sums approach from the other side, and the gap between them is the next omitted magnitude.

## Convergence by Trapping

### Monotone Subsequences

The convergence test for alternating series is not a [comparison test](/theorems/173) in the usual sense. The absolute values may form a divergent positive series. Instead, the argument is an order argument: one monotone subsequence rises, another monotone subsequence falls, and the two are squeezed together.

[quotetheorem:177]

Once convergence is known, the next question is where the limit sits relative to the finite partial sums. Alternating series have a stronger structure than an arbitrary convergent series: the odd and even partial sums form a nested corridor around the limiting value.