Question

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

Answer 1

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`.