Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$
Note : In wolfram alpha show this result and in the same time by ratio test it's not a convince way to judg that is convergent series 
Thank you for any help  

Let $A_N$ be the $N$th partial sum of $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}.$
Let $B_N$ be the $N$th partial sum of $\sum_{n=1}^{\infty}\frac{(-1)^{n}}{2n}.$
Let $C_N$ be the $N$th partial sum of $\sum_{n=1}^{\infty}\frac{(-1)^{n(n-1)/2}}{n}.$
Then $C_{2N} = A_N + B_N.$ By the alternating series test, both $A_N,B_N$ converge. Hence $C_{2N}$ converges to some limit $S$. Now $C_{2N+1}=C_{2N+1}-C_{2N}+C_{2N}$ and $C_{2N+1}-C_{2N}=(\pm 1)/(2N+1) \to 0.$ Thus $C_{2N+1}\to S.$
A sequence whose even terms converge to $S$ and whose odd terms converge to $S$ itself converges to $S.$ Thus $C_N$ converges, which is the desired result.
