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)\frac{\frac7{10n+3\sqrt n}}{\frac7{10n}}=\lim_(n\to\infty)(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).