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

Mar 18, 2018

See below.

#### Explanation:

Considering

${a}_{n} = {\left(5 n\right)}^{3 n} / {\left({5}^{n} + 3\right)}^{n}$

we have

${a}_{n} < {\left(5 n\right)}^{3 n} / {\left({5}^{n}\right)}^{n}$

now we will compare ${a}_{n}$ with $\frac{1}{n} ^ 2$ that we know to converge.

Taking $\ln$

$\ln {a}_{n} < \left(3 n\right) \ln \left(5 + \ln n\right) - {n}^{2} \ln 5$ and as we can easily verify exists a ${n}_{0}$ such that $n > {n}_{0} \Rightarrow \ln {a}_{n} < \ln \left(\frac{1}{n} ^ 2\right)$

As we know, $\ln$ is a strictly increasing function so considering $n > {n}_{0}$

$\ln {a}_{n} < \ln \left(\frac{1}{n} ^ 2\right) \Rightarrow {a}_{n} < \frac{1}{n} ^ 2$

hence the series

${\sum}_{n = 1}^{\infty} {\left(5 n\right)}^{3 n} / {\left({5}^{n} + 3\right)}^{n}$

converges.