Does the series sum_{n=1}^oo (5n)^(3n)/(5^n+3)^n converge or diverge?

1 Answer
Mar 18, 2018

See below.

Explanation:

Considering

a_n = (5n)^(3n)/(5^n+3)^n

we have

a_n < (5n)^(3n)/(5^n)^n

now we will compare a_n with 1/n^2 that we know to converge.

Taking ln

ln a_n < (3n)ln(5+ln n)-n^2 ln 5 and as we can easily verify exists a n_0 such that n > n_0 rArr ln a_n < ln (1/n^2)

As we know, ln is a strictly increasing function so considering n > n_0

lna_n < ln (1/n^2) rArr a_n < 1/n^2

hence the series

sum_{n=1}^oo (5n)^(3n)/(5^n+3)^n

converges.