Find all the parameter values for which the series converges

Question

I need to find all the values of $a$ for which the series beneath converges. $$ \sum_{k=1}^{\infty}\left(e^{\tan\frac{1}{k}-1}\right)^a\ \ \ (*) $$ I did the following: $$ e^{\tan\frac{1}{k}-1}\sim e^{\frac{1}{k}-1}\sim \left(\frac{1}{k}-1\right)+1=\frac{1}{k}\Rightarrow\\ (*)\ \text{converges} \iff \sum_{k=1}^{\infty}\frac{1}{k^a} \text{converges} \Rightarrow\\ (*)\ \text{converges} \iff a > 1 $$ Have I done everything correctly?

Answer 1

We have that

$$\left(e^{\tan\frac{1}{k}-1}\right)^a \to \frac 1{e^a}$$

then the series can't converge.