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

opatovaL 2021-03-07 Answered
Use the Root Test to determine the convergence or divergence of the series.
n=1(n500)n
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Expert Answer

smallq9
Answered 2021-03-08 Author has 106 answers

Given: n=1(n500)n
Root test: Suppose we have the series an, Define L=limn|an|n=limn|an|1n
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,
an=(n500)n
L=limn|(n500)n|1n
=limnn500
=500
=>1
As L>1, by the Root Test the series is divergent.

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Jeffrey Jordon
Answered 2021-12-27 Author has 2064 answers

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