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

1 Answer
Mar 18, 2018

See below.

Explanation:

Considering

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

we have

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

now we will compare a_nan with 1/n^21n2 that we know to converge.

Taking lnln

ln a_n < (3n)ln(5+ln n)-n^2 ln 5lnan<(3n)ln(5+lnn)n2ln5 and as we can easily verify exists a n_0n0 such that n > n_0 rArr ln a_n < ln (1/n^2)n>n0lnan<ln(1n2)

As we know, lnln is a strictly increasing function so considering n > n_0n>n0

lna_n < ln (1/n^2) rArr a_n < 1/n^2lnan<ln(1n2)an<1n2

hence the series

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

converges.