Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.sum_{n=1}^inftyfrac{arctan n}{n^2+1}

beljuA 2020-11-08 Answered

Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.
n=1arctannn2+1

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hesgidiauE
Answered 2020-11-09 Author has 106 answers
To Determine:
Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.
Given: we have n=1tan1nn2+1
Explanation: integral test applies when the function is positive and increasing for nN
we can write it as
1tan1xx2+1dx=lima1atan1xx2+1
let,
u=tan1x
du=1x2+1
when x=1 then u=tan1xu=π4
when x=a then u=tan1a
the integral becomes
limaπ4tan1audu=lima[u22]π4tan1a
=lima12(tan1a)212(π4)2
[limatan1a=tan1=π2]
limaπ4tan1audu=12((π2)2((π4)2))
=12(π24π216)
=3π232
here we see that the series converges.
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Jeffrey Jordon
Answered 2021-12-17 Author has 2070 answers

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