I am trying to study the converge of the series $$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$ But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? Thank you!
$\begingroup$
$\endgroup$
1
-
2$\begingroup$ What do you study it for, if I may ask? $\endgroup$– Fedor PetrovCommented Oct 6 at 11:18
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
By the Dirichlet test it suffices to show that the sums $$\sum_{n=1}^N (-1)^n e^{\sin n}$$ are bounded. This is true - to see this, apply this answer to $f(x) = -e^{\sin(2x-1)} + e^{\sin(2x)}$ for example.
