Given: \(\sum_{n=1}^\infty(\frac{n}{500})^n\)

Root test: Suppose we have the series \(\sum a_n\), Define \(L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}|a_n|^{\frac{1}{n}}\)

Then,

1) If L

2) If L>1, the series is divergent.

3) If L=1, the series may be divergent, conditionally convergent, or absolutely convergent.

Now consider,

\(a_n=(\frac{n}{500})^n\)

\(L=\lim_{n\to\infty}|(\frac{n}{500})^n|^{\frac1n}\)

\(=\lim_{n\to\infty}\frac{n}{500}\)

\(=\frac{\infty}{500}\)

\(=\infty>1\)

As L>1, by the Root Test the series is divergent.

Root test: Suppose we have the series \(\sum a_n\), Define \(L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}|a_n|^{\frac{1}{n}}\)

Then,

1) If L

2) If L>1, the series is divergent.

3) If L=1, the series may be divergent, conditionally convergent, or absolutely convergent.

Now consider,

\(a_n=(\frac{n}{500})^n\)

\(L=\lim_{n\to\infty}|(\frac{n}{500})^n|^{\frac1n}\)

\(=\lim_{n\to\infty}\frac{n}{500}\)

\(=\frac{\infty}{500}\)

\(=\infty>1\)

As L>1, by the Root Test the series is divergent.