Does $\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\right)$ converges conditionally? I'm trying to understand whether the following series converges absolutely, conditionally or diverges. 
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\right)
$$
I think that it converges conditionally. But it's getting complicated for me to show that it does not converge absolutely. I've already tried the limit comparison test with $b_n = \frac{1}{\sqrt{n}}$ but the limit results in $0$. I've also tried the integral test, as the conditions for the test hold, but the integral became very complicated. 
Is there an easier way?

I think the integral test works.
The taylor series of $\cos(x)$ is
$$\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}x^{2k}$$
Thus, $$1-\cos(\frac1{\sqrt{n}})= -\sum^\infty_{k=1} \frac{(-1)^k}{(2k)!}\frac1{n^{k}}$$
$$-\int^\infty_1 \sum^\infty_{k=1} \frac{(-1)^k}{(2k)!}\frac1{x^{k}}dx=-\int^\infty_1 \sum^\infty_{k=2} \frac{(-1)^k}{(2k)!}\frac1{x^{k}}dx+\int^\infty_1\frac{1}{2x}dx=C+\infty$$ where $C$ is finite.
Therefore the sum does not converge absolutely.
It is an elementary exercise to prove $C$ is finite.:)
