Show that $\sum_{n=2}^{\infty}\frac{(-1)^{n}}{\ln n} \left(1+\tan\frac{1}{n} \right)^{2019}$ converges.

Question

Show that $$\sum_{n=2}^{\infty}\frac{(-1)^{n}}{\ln n} \left(1+\tan\frac{1}{n} \right)^{2019}$$ converges.

I wanted to use the alternating series test with $b_{n} = \frac{(1+\tan\frac{1}{n})^{2019}}{\ln n}$ but so far this hasn't worked out for me, so I feel like this is the wrong test. Do you know which test would work?

Answer 1

The alternating series test is the correct one. Which part of it isn't working for you?