Prove that the series ∞Xn=12nn!is convergent.1

To prove that the series ∑ (1/2n*n!) from n=1 to ∞ is convergent, we can use the Ratio Test.

The Ratio Test states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of a series is less than 1, then the series is absolutely convergent.

The nth term of our series is (1/2nn!). The (n+1)th term is (1/2^(n+1)(n+1)!).

The ratio of the (n+1)th term to the nth term is [(1/2^(n+1)(n+1)!)] / [(1/2nn!)].

Simplify this ratio to get (1/2(n+1)).

Take the limit as n approaches infinity of this ratio. The limit is 0, which is less than 1.

Therefore, by the Ratio Test, the series ∑ (1/2n*n!) from n=1 to ∞ is absolutely convergent.