How to determine that the series
$$\sum_{n=1}^{\infty}\arctan(e^{-n})$$
is convergent by using ratio test?
I tried to compute the ratio of
$$\lim_{n\to \infty}\frac{\arctan(e^{-n-1})}{\arctan(e^{-n})}$$
But I do not know what is the result of that…
Best Answer
Are you able to use that for small positive $x$, $\frac{1}{2}x<\arctan(x)<x$?
If so, then $$0<\frac{\arctan(e^{-n-1})}{\arctan(e^{-n})}<\frac{e^{-n-1}}{\arctan(e^{-n})}<\frac{e^{-n-1}}{\frac12e^{-n}}=\frac{2}{e}<1$$