# Use the Root Test to determine the convergence or divergence of the series. sum_{n=1}^infty(frac{n}{500})^n

Use the Root Test to determine the convergence or divergence of the series.
$\sum _{n=1}^{\mathrm{\infty }}\left(\frac{n}{500}{\right)}^{n}$
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Given: $\sum _{n=1}^{\mathrm{\infty }}\left(\frac{n}{500}{\right)}^{n}$
Root test: Suppose we have the series $\sum {a}_{n}$, Define $L=\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{|{a}_{n}|}=\underset{n\to \mathrm{\infty }}{lim}|{a}_{n}{|}^{\frac{1}{n}}$
Then,
1) If $L<1$, the series is absolutely convergent (and hence convergent).
2) If $L>1$, the series is divergent.
3) If $L=1$, the series may be divergent, conditionally convergent, or absolutely convergent.
Now consider,
${a}_{n}=\left(\frac{n}{500}{\right)}^{n}$
$L=\underset{n\to \mathrm{\infty }}{lim}|\left(\frac{n}{500}{\right)}^{n}{|}^{\frac{1}{n}}$
$=\underset{n\to \mathrm{\infty }}{lim}\frac{n}{500}$
$=\frac{\mathrm{\infty }}{500}$
$=\mathrm{\infty }>1$
As $L>1$, by the Root Test the series is divergent.

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