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

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

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Answer 1 · 2015-10-21T01:15:06

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

1 Answer

Explanation:

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