Question

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

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asked 2021-03-07
Use the Root Test to determine the convergence or divergence of the series.
\(\sum_{n=1}^\infty(\frac{n}{500})^n\)

Answers (1)

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 \(L<1\), 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,
\(a_n=(\frac{n}{500})^n\)
\(L=\lim_{n\to\infty}|(\frac{n}{500})^n|^{\frac1n}\)
\(=\lim_{n\to\infty}\frac{n}{500}\)
\(=\frac{\infty}{500}\)
\(=\infty>1\)
As \(L>1\), by the Root Test the series is divergent.

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