Determining if a Series if Absolutely Convergent
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Determining if a Series if Absolutely Convergent

[From: ] [author: ] [Date: 11-11-19] [Hit: ]
Use the Root Test.L = 1/4.Hope this helped.-No problem. Math is a passion of mine, anytime.......
Looking for assistance (some step by steps, hopefully, not just the answer) with the following homework problem:

Determine whether the series is absolutely convergent:

∑ ((n^2 + 1) / (2n^2 + 1))^2n, where n = 1, to infinity

It's giving me fits so thanks so much in advance for any assistance!

-
oo
∑ ((n^2 + 1) / (2n^2 + 1))^2n
n=1

It seems like the whole fraction is raised to the 2n.

Use the Root Test.

L = lim n-->oo | [(n^2 + 1) / (2n^2 + 1)]^(2n) | ^(1/n)
L = lim n-->oo | (n^2 + 1)^2 / (2n^2 + 1)^2 |
L = lim n-->oo | (n^4 + 2n^2 + 1) / (4n^4 + 4n^2 + 1) |
L = lim n-->oo | (1 + 2/n^2 + 1/n^4) / (4 + 4/n^2 + 1/n^4) |
L = 1/4.

Since 1/4 < 1, the series converges absolutely and hence converges.

Hope this helped.

-
No problem. Math is a passion of mine, anytime.

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