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Find the sum Sigma(r=1)^(oo) (r)/(r^4+1/...

Find the sum `Sigma_(r=1)^(oo) (r)/(r^4+1/4)`

Text Solution

Verified by Experts

The correct Answer is:
`840/841`
`T_(r)=r/(r^(4)+1/4)=r/(r^(4)+r^(2)+1/4-r^(2))=r/((r^(2)+1/2)^(2)-r^(2))`
`=1/2[(2r)/((r^(2)+r+1/2)(r^(2)-r+1/2))]`
`=1/2[((r^(2)+r+1/2)-(r^(2)-r+1/2))/((r^(2)+r+1/2)(r^(2)-r+1/2))]`
`=1/2[1/(r^(2)-r+1/2)-1/(r^(2)+r+1/2)]`
`=[1/(2r^(2)-2r+1)-1/(2r^(2)+2r+1)]`
`thereforesum_(r=1)^(20)T_(r)=sum_(r=1)^(20)[1/(2r^(2)-2r+1)-1/(2r^(2)+2r+1)]`
`=1/(2(1)^(2)-2(1)+1)-1/(2(20)^(2)+2(20)+1)`
`=1-1/841-840/841`
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