Explained: "Use the Root Test to determine the convergence..."

opatovaL

Answered question

2021-03-07

Use the Root Test to determine the convergence or divergence of the series.

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

Answer & Explanation

smallq9

Skilled2021-03-08Added 106 answers

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

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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