Question

Determine if the series converges or diverges

Answer 1

Compare to the series

$\displaystyle\sum_{n=1}^\infty\frac7{10n}$

which is divergent. We have

$\displaystyle\lim_{n\to\infty}\dfrac{\frac7{10n+3\sqrt n}}{\frac7{10n}}=\lim_{n\to\infty}\dfrac{10n}{10n+3\sqrt n}=1$

Because this limit is finite and positive, the original series must also be divergent (by the limit comparison test).