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Problem 7 Easy Difficulty

Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 0}^{\infty} \frac {x^n}{n!} $

Answer

$$R=\infty \text { and } I=(-\infty, \infty)$$

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Video Transcript

okay for this problem. Our radius of convergence are we have limited as n goes to infinity, absolute value of a N over and plus one. And this time that's going to end up looking like in plus one factorial divided by in factorial. Remember the factorial function, in fact or religious like one times two times, three times four times Thought that thought Although I have two times in, there's going to be lots of cancellations happening here. Everything will cancel except for in plus one up top. Okay, to the N plus one will stay there and this limit is clearly infinity. It's our radius of convergence is infinity and typically with the interval of convergence, you would wantto check to see if you know, plus or minus our worked. But in this case, it's ours, just infinity. So our interval of convergence is still just going to be everything all the way. Everything between minus infinity and infinity

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