If possible, use the ratio test to determine whether the series

converges or diverges.

We need to look at the limit of

as n approaches ∞.

The terms a_{n + 1} and a_{n} are both fractions:

To avoid fractions within fractions, we write a_{n + 1} multiplied by the reciprocal of a_{n}.

Since (n + 1)! = (n + 1)n! and since 2^{n + 1} = 2^{n}2^{1} we can cancel a lot of things:

The ratio test says to look at the limit of the ratios

This limit diverges, so by the ratio test the series diverges also.