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?

Note that 
$$1-\cos\left(\frac{1}{\sqrt{n}}\right)= \frac1{2n}+O\left(\frac1{n\sqrt n}\right)$$
then the given series doesn’t converge absolutely by limit comparison test with $\sum \frac1{n}$ and then
$$
\sum _{n=1}^{\infty }\left(-1\right)^{n+1}\left(1-\cos\left(\frac{1}{\sqrt{n}}\right)\right)=\sum _{n=1}^{\infty }\left(-1\right)^{n+1} \frac1{2n}+\sum _{n=1}^{\infty }\left(-1\right)^{n+1} O\left(\frac1{n\sqrt n}\right)
$$
which converges since


*

*$\sum _{n=1}^{\infty }\left(-1\right)^{n+1} \frac{1}{2n}$ converges by alternating series test

*$\sum _{n=1}^{\infty }\left(-1\right)^{n+1} O\left(\frac1{n\sqrt n}\right)$ converges absolutely by limit comparison test with $\sum \frac1{n\sqrt n}$
