Series convergence $(-1)^n n\tan\frac{1}{n}$ I tried to prove the convergence of this series with the ratio and root test but neither worked. I also couldn't find another series to compare.
I understand that $\tan\frac{1}{n}$ is decreasing  and the partial sums too.
I have no idea what to try now
$$\sum _{n=1}^{\infty }(-1)^n n\tan\frac{1}{n}.$$

$$n\tan\left(\dfrac1n\right)=\frac1{\cos\left(\dfrac1n\right)}\frac{\sin\left(\dfrac1n\right)}{\dfrac 1n}$$ shows that the general term does not tend to zero.
