Q: How to get an approximative behaviour of an alternating sum I have an alternating sum and am looking for its asymptotic behaviour. Due to the change of signs there occure a lot of cancellations between the different terms. However the terms are all different, so no terms cancel exactly. 
Is there still a method that can be used to find the dominating behaviour of the sum? 
Thanks

A: You can group pairs of terms together and study the new sum.
For example, if your series is $\sum_{n=1}^{\infty}(-1)^n/n$, you can group the $(2n)^{th}$ and $(2n+1)^{th}$ terms together. They nearly cancel out to give $1/(2n)(2n+1)$, so
$$
\sum_{n=1}^{\infty}\frac{(-1)^n}{n}=1+\sum_{n=1}^{\infty}\frac{1}{2n(2n+1)},
$$
and so the odd partial sums of the first series have the same asymptotic behavior as the partial sums on the right, whereas the even partial sums have a (slightly different) asymptotic behavior.
