Exponential series/Real/Absolute convergence/Fact/Proof

Proof

For ${}x=0$, the statement is clear. Else, we consider the fraction

${}\vert {\frac {\frac {x^{n+1}}{(n+1)!}}{\frac {x^{n}}{n!}}}\vert =\vert {\frac {x}{n+1}}\vert ={\frac {\vert {x}\vert }{n+1}}\,.$

This is, for ${}n\geq 2\vert {x}\vert$, smaller than ${}1/2$. By the ratio test, we get convergence.

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