Question

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

Answer 1

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.