How do you use the ratio test to test the convergence of the series #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.