Does the sum of the series Xn between 1 and infinity converge?

Question

When Xn = (n^(-2))*(log(n))^4
How do you work it out?
Thanks.

Answer 1

Write it as (log n)^4/n^1/2 * 1/n^3/2
The factor (log n)^4/n^1/2 has limit 0 ( you can prove it using L´Hôpital) so from a certain in advance it is <1
So (log n)^4/n^2<1/n^3/2 which is a p series with p>1 convergent and so the given series is convergent