How do you use the ratio test to test the convergence of the series ∑ (2n^2)/(n!) from n=1 to infinity?

1 Answer
Bill K.
Oct 21, 2015

The infinite series converges (see below).

Explanation:

Let a_{n}=(2n^2)/(n!). Then |a_{n+1}|/|a_{n}|=(2(n+1)^2)/((n+1)!) * (n!)/(2n^2)=((n+1)^2)/(n^2(n+1))=(n+1)/n^2 for all positive integers n.

Since lim_{n->infty}|a_{n+1}|/|a_{n}|=lim_{n->infty}(n+1)/n^2=0<1, it follows that the infinite series sum_{n=1}^{infty}(2n^2)/(n!) converges by the Ratio Test .

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