What is the radius of convergence of the series $sum_(n=0)^oo(x^n)/(n!)$ ?

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Answer 1 · 2014-09-22T08:10:21

The radius of convergence is $infty$ .

Let us look at some details.

Let $a_n=x^n/{n!}$ . $Rightarrow a_{n+1}=x^{n+1}/{(n+1)!}$

By Ratio Test,

$lim_{n to infty}|{a_{n+1}}/{a_n}| =lim_{n to infty}|{x^{n+1}/{(n+1)!}}/{x^n/{n!}}| =lim_{n to infty}|x/{n+1}|=0<1$

Since $lim_{n to infty}|{a_{n+1}}/{a_n}|<1$ regardless of the value of $x$ , thr radius of convergence is $infty$ .

What is the radius of convergence of the series sum_(n=0)^oo(x^n)/(n!)sum_(n=0)^oo(x^n)/(n!)?