Limit of Ratio Test Is $0$ (Convergence or Divergence?)
I have to check to see if the sum of the series of the function $\frac{x^n}{n!}$ converges or diverges.
I decided to use the ratio test where I took the limit as $n$ approaches infinity of the ratio between a term ($a_n$) and its previous term ($a_{n-1})$. The limit I get is $0$. I know that if the limit approaches a value less than $1$, then I can say that the series converges. But here although $0 < 1$, I'm not sure if approaching a ratio of $0$ between terms will prove the function converges.
$\endgroup$1 Answer
$\begingroup$You have shown that the limit of the ratio is $0$ for any $x$, so we can say that the function converges for all $x$ by the ratio test. (This is the exponential function.)
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