How do you use the ratio test to test the convergence of the series $\infty \sum n = 1 \frac{{(x + 1)}^{n}}{n!}$ $sum_(n=1)^oo((x+1)^n) / (n!)$ ?

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Answer 1 · 2017-05-04T00:55:40

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

If $L<1$ then absolutely convergent
If $L=1$ inconclusive
If $L>1$

Given: $A_n = ((x+1)^n)/(n!)$

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

$A_(n+1)/A_n= (((x+1)^(n+1))/((n+1)!))/(((x+1)^n)/(n!)) = (((x+1)^(n+1))/(n+1))/(((x+1)^n)/1)= (x+1)/(n+1)$

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

Absolutely convergent.

How do you use the ratio test to test the convergence of the series sum_(n=1)^oo((x+1)^n) / (n!)∞∑n=1(x+1)nn!sum_(n=1)^oo((x+1)^n) / (n!) ?