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

Options

A. 0
B. `(1)/(2)`
C. 2
D. 4

Answer

C

Explanation

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))]`
`lim_(n to oo)S_(n)=lim_(n to oo)S_(n)sum_(n=1)^(n)T_(r)`
`lim_(n to oo)sum_(n=1)^(n)2((1)/(2^(2)-2r+1)-(1)/(2^(2)+2r+1))`
`2lim_(n to oo)(1-(1)/(2^(2)+2r+1))=2(1-0)=2`.