Alternating series theorem

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This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

Basic statement

Consider a series of the form:

$a_{1}+a_{2}+\dots +a_{n}+\dots =\sum _{k=1}^{\infty }a_{k}$

Suppose the following three conditions hold for the series:

Alternating signs: All the $a_{k}$ s are nonzero and the sign of $a_{k+1}$ is opposite the sign of $a_{k}$ for all $k$ .
Monotonically decreasing (i.e., non-increasing) in magnitude: $|a_{k}|\geq |a_{k+1}|$ for all $k$ .
Terms approach zero: $\lim _{k\to \infty }a_{k}=0$ . This is equivalent to saying that $\lim _{k\to \infty }|a_{k}|=0$

Then the series is a convergent series. It may be an absolutely convergent series or a conditionally convergent series, depending on whether the series of the absolute values of its terms converges.

Concrete version assuming first term is positive

Suppose the basic statement holds and we additionally assume that $a_{1}$ is positive. Then, $a_{k}$ is positive for odd $k$ and $a_{k}$ is negative for even $k$ . The alternating series theorem states, concretely, that:

The partial sums for the first $k$ terms for $k$ odd form a monotonically decreasing sequence bounded from below, hence has a limit.
The partial sums for the first $k$ terms for $k$ even form a monotonically increasing sequence bounded from above, hence has a limit.
Both the limits are equal.

Concrete version assuming first term is negative

Suppose the basic statement holds and we additionally assume that $a_{1}$ is positive. Then, $a_{k}$ is negative for odd $k$ and $a_{k}$ is positive for even $k$ . The alternating series theorem states, concretely, that:

The partial sums for the first $k$ terms for $k$ odd form a monotonically increasing sequence bounded from above, hence has a limit.
The partial sums for the first $k$ terms for $k$ even form a monotonically decreasing sequence bounded from below, hence has a limit.
Both the limits are equal.

Eventually variation

The alternating series theorem can be generalized as follows: we do not require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series eventually be alternating and eventually the terms be monotonically decreasing in magnitude.

Significance

Significance for showing convergence

The alternating series theorem is widely used in showing the convergence of series. Specifically, it helps show the convergence of series of the form $\sum (-1)^{k}b_{k}$ where $b_{k}$ (eventually) have constant sign and are monotonically decreasing in magnitude. A couple of applications are below:

The signed version of the degree difference test asks for conditions for a rational function $\sum (-1)^{k}{\frac {p(k)}{q(k)}}$ to converge. Based on the alternating series theorem, we obtain that convergence occurs if and only if $\operatorname {deg} (q)-\operatorname {deg} (p)>0$ . Moreover, the convergence is absolute if the degree difference is greater than 1 and conditional if the degree difference is greater than 0 and less than or equal to 1.
The alternating series theorem plays a key role, either directly or via the degree difference test, in the rules for determining interval of convergence. Specifically, it helps determine for certain power series whether endpoints are included in the interval of convergence.

Significance for approximate computation of sums of series

The concrete version of the alternating series theorem can be used to compute upper and lower bounds for the sums of alternating series. Specifically, all partial sums ending at negative terms give lower bounds and all partial sums ending at positive terms give upper bounds.

In general, alternating series that are absolutely convergent have fairly rapid convergence, and this gives a very effective method for approximating the sum. Alternating series that are not absolutely convergent have fairly slow convergence, and the method here can be quite slow for approximating the sum (i.e., we need to add a lot of terms to achieve a desired level of accuracy).

Related facts

Significance of each condition

Alternating series theorem fails if signs are not strictly alternating: If we replace the strict alternation condition on signs by simply saying that the sign switches between positive and negative infinitely often, that is not sufficient to guarantee convergence of the alternating series. Note that it may still happen that the series converges, but there is no guarantee by the theorem.
Alternating series theorem fails if terms are not monotonically decreasing in magnitude: If we drop the condition that the magnitude of the terms is monotonically decreasing, then the series need not converge. Note that it may still happen to converge, but there is no guarantee by the theorem.
Analogue of alternating series theorem if magnitude of terms approaches a positive number: The alternating series theorem fails if the magnitude of terms does not approach zero. However, we can formulate an analogous theorem which says that the difference between the limit superior and limit inferior of the partial sums is precisely that positive number limit.