Find the sum of the convergent series. sum_{n=1}^inftyfrac{1}{9n^2+3n-2}

Marvin Mccormick 2020-10-25 Answered
Find the sum of the convergent series.
n=119n2+3n2
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Expert Answer

delilnaT
Answered 2020-10-26 Author has 94 answers
Solution:
Consider the given series is n=119n2+3n2
Factorise denominator 9n2+3n2=9n2+6n3n2=(3n1)(3n+2)
therefore, the series will be like n=11(3n1)(3n+2)
or, n=11(3n1)(3n+2)=n=119(n13)(n+23)
This is the form of Telescoping series of an=n13 and an+1=n+23
By telescoping series result n=1(1n+k)(1n+k+1)=1k+1
Applying these result to given series,
n=119(n13)(n+23)=19n=11(n13)(n+23)=19×123=318=16
Hence, the sum of the given convergent series is 16
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Jeffrey Jordon
Answered 2021-12-25 Author has 2047 answers

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