$\sum_{i=0}^{\infty} e^{-\sqrt n}z^n $
I tried to find the radius of convergence of a power series.. is this equation a geometric series?
or would it be easier to do a ratio test and
$\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L$ Then radius of convergence $R=1/L$?
I'm got it converges to $0$, not confident about it though.
Best Answer
But you can use that ratio test:\begin{align}\lim_{n\to\infty}\frac{e^{-\sqrt{n+1}}}{e^{-\sqrt n}}&=\lim_{n\to\infty}e^{\sqrt n-\sqrt{n+1}}\\&=e^{\lim_{n\to\infty}\sqrt n-\sqrt{n+1}}\\&=e^0\\&=1.\end{align}Therefore, the radius of convergence is $1$.