Does $sum_(n=1)^oo 3^n/(n!)$ converge ?

Question

$sum_(n=1)^oo 3^n/(n!)$

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Answer 1 · 2017-06-04T16:01:29

Yes. By the ratio test

$lim_(nrarroo)|A_(n+1)/A_n|$

$A_n = 3^n/(n!)$

$A_(n+1) = (3/(n+1))3^n/(n!)$

$lim_(nrarroo)|3/(n+1)| = 0$

I asked WolframAlpha; it confirmed my answer and gave me a value:

$sum_(n=1)^oo 3^n/(n!) = e^3-1$