Many of the series convergence tests that have been introduced so far are stated with the assumption that all terms in the series are nonnegative. Indeed, this condition is assumed in the Integral Test, Ratio Test, Root Test, Comparison Test and Limit Comparison Test. In this section, we study series whose terms are not assumed to be strictly positive. In particular, we are interested in series whose terms alternate between positive and negative (aptly named alternating series). It turns out that there is a powerful test for determining that a series of this form converges.

Compared to our convergence tests for series with strictly positive terms, this test is strikingly simple. Let us examine why it might be true by considering the partial sums of the alternating harmonic series. The first few partial sums with odd index are given by $$\begin{align*} s_1 &= 1 \\ s_3 &= 1 - \frac {1}{2} + \frac {1}{3} = s_1 - \left (\frac {1}{2} - \frac {1}{3}\right ) = \frac {5}{6} \\ s_5 &= s_3 - \frac {1}{4} + \frac {1}{5} = s_3 - \left ( \frac {1}{4} - \frac {1}{5}\right ) = \frac {47}{60}. \end{align*}$$

Note that a general odd partial sum is of the form

and the quantity in the parentheses is positive. We conclude that:

Moreover, the sequence of odd partial sums is bounded below by zero, since

and each quantity in parentheses is positive.

Next consider the sequence $$\begin{align*} s_2 &= 1 - \frac {1}{2} = \frac {1}{2} \\ s_4 &= 1- \frac {1}{2} + \frac {1}{3} - \frac {1}{4} = \frac {7}{12} \\ s_6 &= 1 - \frac {1}{2} + \frac {1}{3} - \frac {1}{4} + \frac {1}{5} - \frac {1}{6} = \frac {37}{60} \\ &\vdots \end{align*}$$

of even partial sums.

Finally, we use

and the fact that the limits of the sequences $\{s_{2n+1}\}$ and $\{s_{2n}\}$ are finite to conclude that

It follows that the alternating harmonic series converges.

With slight modification, the argument given above can be used to prove that the Alternating Series Test holds in general.