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Find the sum Sigma(n=1)^(oo)(3n^2+1)/((n...

Find the sum `Sigma_(n=1)^(oo)(3n^2+1)/((n^2-1)^3)`

Text Solution

Verified by Experts

The correct Answer is:
`9/16`
`T_(n)=(3n^(2)+1)/((n^(2)-1)^(3))=1/2(6n^(2)+2)/((n^(2)-1)^(3))`
`=1/2(((n+1)^(3)-(n-1)^(3))/((n+1)^(3)(n-1)^(3)))`
`=1/2(1/((n-1)^(3))-1/(n+1)^(3))`
`thereforeS=1/2[(1/1^(3)-1/3^(3))+(1/2^(3)-1/3^(3))+(1/2^(3)-1/4^(3))+(1/3^(3)-1/5^(3))`
`+(1/4^(3)-1/6^(3))+...]`
`=1/2(1+1/8)=9/16`
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