I want to determine if the series $ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $ converge/diverge. the sequence in the denominator is not monotinic, so I cant use Dirichlet's or Abel's tests. My intuition is that this series converge, becuase its looks close to $ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{n} $ but im not sure how to prove. Any ideas will help, thanks.
Determine whether or not infinite series $ \sum_{n=2}^{\infty}\frac{\left(-1\right)^{n}}{\left(-1\right)^{n}+n} $ converge
calculuslimitssequences-and-series
Best Answer
Let
$$s_n=\sum_{k=2}^n\frac{(-1)^k}{(-1)^k+k}=\frac13-\frac12+\frac15-\frac14+\ldots+\frac{(-1)^n}{(-1)^n+n}$$
and
$$s_n'=\sum_{k=2}^n\frac{(-1)^{k+1}}k=-\frac12+\frac13-\frac14+\frac15+\ldots+\frac{(-1)^{n+1}}n\;.$$
Show that $s_{2n+1}=s_{2n+1}'$ and $s_{2n}=s_{2n+1}'+\frac1{2n}$ for $n\ge 1$. Use this or the fact that $s_{2n}=s_{2n}'+\frac1{2n}+\frac1{2n+1}$ to show that $\lim_\limits{n\to\infty}|s_n-s_n'|=0$, and therefore $\lim_\limits{n\to\infty}s_n=\lim_\limits{n\to\infty}s_n'$.