Apply the ratio test to the series. Determine if the series converges, diverges, or requires a different test.
For the series, find
We have . So
(the last step is from looking at the leading terms in the numerator and denominator).
the Ratio Test says this series converges.
L = 9 > 1
the ratio test says this series diverges.
Applying the ratio test gives us
Getting L = 1 means we need a different test. Thankfully the integral test, will be useful for this series.
Be careful with the parentheses and factorials here.
(since we have an n2 term in the numerator and only a constant in the denominator). The ratio test says this series diverges.
Since we take the absolute value of the ratio, we don't have to worry about the negative sign.
L = 0 < 1
the ratio test says this series converges.
Make it rain.