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Determine if the series converges or diverges

Determine if the series converges or diverges-example-1

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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).
User Ben Thompson
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