If $a_n>0$, then the alternating series $\sum_{n=1}^\infin (-1)^n a_n$ or $\sum_{n=1}^\infin (-1)^{n+1} a_n$ converges if both of the following conditions are satisfied:

1. $\lim_{n→\infin} a_n=0$
2. ${a_n}$ is a decreasing (Non-increasing) sequence; that is $a_{n+1}\le a_n$ for all n>k, for some $k\in \mathbb Z$ (integer)
   1. eventually decreasing

Error

Suppose an alternating series satisfies the conditions of the AST,namely that $\lim_{n→\infin} a_n =0$ and ${a_n}$ is not increasing. If the series has a sum S, then $|R_n| =|S-S_n|\le a_{n+1}$ ,where $S$ is the nth partial sum of the series.

In other words, if an alternating series satisfies the conditions of the AST, you can approximate the sum of the series by using the nth partial sum, $S_n$, and your error will have an absolute value no greater than the first term left off, $a_{n+1}$. This means $[S_n-R_n,S_n+R_n]$.

• different from integral test, where $0\le R_n \le \int_n^\infin f(x)dx$, $a_{n+1}$ is the error that could be negative or positive

• proof