# Which of the series, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) sum_{n=1}^inftyfrac{3}{sqrt n}

Which of the series, and which diverge?
Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
$\sum _{n=1}^{\mathrm{\infty }}\frac{3}{\sqrt{n}}$
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Tasneem Almond
Given:
$\sum _{n=1}^{\mathrm{\infty }}\frac{3}{\sqrt{n}}$
To find the series is convergent or divergent use the ratio test for series.
$L=\underset{n\to \mathrm{\infty }}{lim}|\frac{{a}_{n+1}}{{a}_{n}}|$
If L<1, then the series is convergent.
If L>1, Then the series is divergent.
If L=1, Then the ratio test fails.
Apply ratio test
$|\frac{{a}_{n+1}}{{a}_{n}}|=\frac{\frac{3}{\sqrt{n+1}}}{\frac{3}{\sqrt{n}}}$
$=\frac{\sqrt{\frac{n}{n}}}{\sqrt{1+\frac{1}{n}}}$
$=1$
the ratio test fails
Jeffrey Jordon