Question

determine whether the series converges or diverges. I am stuck on these two problems. Any help is appreciated.

infinity
sigma (ln n)/n
n=2

infinity
sigma (5n^3-3n)/[(n^2)(n+2)(n^2+5)]
n=1

Answer 1

The first summation diverges because the terms are asymptoticaly larger than 1/n and the summation of 1/n diverges.

The second summation converges because asymptotically the terms become proportional to 1/n^2 and the summation of 1/n^2 converges.