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 an + 1 and an are both fractions:
To avoid fractions within fractions, we write an + 1 multiplied by the reciprocal of an.
Since (n + 1)! = (n + 1)n! and since 2n + 1 = 2n21 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.