TITLE: Calculating the limit of the sum
QUESTION [0 upvotes]: I would like to receive some help with the next problem:
I'm trying to calculate $\lim_{n \to \infty} \sum_{k = 0}^n \frac{x^k}{k!}$, $x \in \mathbb{R}$. I know that $\lim_{n \to \infty} \frac{x^n}{n!} = 0$, but i don't know if i can use it in this problem, because i don't think i can put limes in sum, like you can do with roots or integrals.  
I'm apologizing, but I'm stuck at the beginning.
Please, could you give me some hints about how to solve this problem?

REPLY [1 votes]: To evaluate $\sum_{k=0}^\infty\frac{x^k}{k!}$, note it solves $y^\prime=y,\,y(0)=1$ so the series is $e^x$. Your original strategy would never work because it only tells you the terms $\to 0$, which is true in any convergent series.
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