An alternating series is a series that alternates between positive and negative terms. Alternating series are in one of these two forms, where $b_n$ is positive:

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

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

alternating series test

If the alternating series $\displaystyle\sum_{n=1}^\infty (-1)^{n-1} b_n$ satisfies:

1. $\displaystyle\lim_{n\to\infty}b_n = 0$
2. $b_{n+1} \le b_n$

Then $\displaystyle\sum_{n=1}^\infty (-1)^{n-1} b_n$ converges.

example: alternating harmonic series

<aside>

Consider the series $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n} = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$ Does this diverge or converge?

</aside>

We can use the alternating series test here:

1. Condition 1 ($\displaystyle\lim_{n\to\infty}b_n$ must equal 0):$\displaystyle\lim_{n\to\infty}b_n = \lim_{n\to\infty}\frac{1}{n} = 0$
2. Condition 2 ($b_{n+1} \le b_n$): $\frac{1}{n+1}\le\frac{1}{n}$.

Therefore, the series converges. (In fact, it converges to $\ln(2)$.)

another example

<aside>

Does the series $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n+1}$ converge or diverge?

</aside>

1. Condition 1 ($\displaystyle\lim_{n\to\infty}b_n$ must equal 0):$\displaystyle\lim_{n\to\infty}b_n = \lim_{n\to\infty}\frac{1}{2n+1} = 0$
2. Condition 2 ($b_{n+1} \le b_n$): $\frac{1}{2(n+1)+1}\le\frac{1}{2n+1}$ (since $2(n+1)+1 > 2n+1$).

Therefore, the series converges.

more examples