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

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

2021-03-08
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.

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