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S(n) be the sum of n terms of the series...

`S_(n)` be the sum of n terms of the series `(8)/(5)+(16)/(65)+(24)/(325)+"......"`
The value of `S_(8)`, is

A

`(288)/(145)`

B

`(1088)/(545)`

C

`(81)/(41)`

D

`(107)/(245)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `S_n=(8)/(5)+(16)/(65)+(24)/(325)+"......"`
`T_(r)=(8r)/(4r^(4)+1)=(8r)/((2^(2)+2r+1)(2^(2)-2r+1))`
`=2[((2^(2)+2r+1)-(2^(2)-2r+1))/((2^(2)+2r+1)(2^(2)-2r+1))]`
`=2[(1)/((2^(2)-2r+1))-(1)/((2^(2)+2r+1))]`
`S_(8)=sum_(r=1)^(8)T_(r)=2sum_(r=1)^(8)((1)/(2^(2)-2r+1)-(1)/(2^(2)+2r+1))`
`=2(1-(1)/(2(8)^(2)+2(8)+1))=2(1-(1)/(145))=(288)/(145)`
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