Answer to Question #161769 in Calculus for Vishal

Question #161769

Show that ∞n=0 (x^n)/(n!)converges for all x ∈ R.


1
Expert's answer
2021-02-24T06:40:00-0500

Using the ration test, the radius of convergence of the series n=0+xnn!\displaystyle \sum_{n=0}^{+\infty}\frac{x^n}{n!} is

r=limncncn+1=limn1n!1(n+1)!=limnn+1=+.r=\lim_{n\rightarrow\infty}\left|\frac{c_n}{c_{n+1}}\right|=\lim_{n\rightarrow\infty}\left|\frac{\frac{1}{n!}}{\frac{1}{(n+1)!}}\right|=\lim_{n\rightarrow\infty}n+1=+\infty.


That is, for all xRx\in\mathbb{R}, the series is convergent.


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