# Find whether the series diverges and its sum: \sum_{n=1}^\infty(-1)^{n+1}\frac{3}{5^n}

Find whether the series diverges and its sum:
$\sum _{n=1}^{\mathrm{\infty }}{\left(-1\right)}^{n+1}\frac{3}{{5}^{n}}$
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Florence Pittman

Notice that
${\left(-1\right)}^{n+1}\frac{3}{{5}^{n}}=-3\frac{{\left(-1\right)}^{n}}{{5}^{n}}=-3{\left(\frac{-1}{5}\right)}^{n}$
Since $\sum _{k=1}^{\mathrm{\infty }}a{r}^{k}=\frac{ar}{1-r}$ (if $|r|<1$)
$\sum _{n=1}^{\mathrm{\infty }}-3{\left(\frac{-1}{5}\right)}^{n}=\frac{-3\cdot \frac{-1}{5}}{1-\frac{-1}{5}}=\frac{\frac{3}{5}}{\frac{6}{5}}=\frac{1}{2}$
and the sum converges because $|\frac{-1}{5}|=\frac{1}{5}<1$

pseudoenergy34