Let `S` denote sum of the series `3/(2^3)+4/(2^4 .3)+5/(2^6 .3)+6/(2^7 .5)+oo` Then the value of `S^(-1)` is __________.

Answer

2

Explanation

Let `S=sum_(r=1)^(oo)(r=2)/(2^(r+1)cdotrcdot(r+1))`
`=sum_(r=1)^(oo)(2(r+1)-r)/(2^(r+1)cdotrcdotr(r+1))`
`=sum_(r=1)^(oo)1/(2^(r+1))(2/r-1/(r+1))`
`=sum_(r=1)^(oo)(1/(2^(r )cdotr)-1/(2^(r+1)(r+1)))`
`=lim_(ntooo)[(1/(2^(1)xx1)-1/(2^(2)xx2))+(1/(2^(2)xx2)-1/(2^(3)xx3))]+(1/(2^(3)xx3)-1/(2^(4)xx4))+....+(1/(2^(n)cdotn)-1/(2^(n+1)cdot(n+1)))`
`=lim_(ntooo)(1/2-1/(2^(n+1)(n+1)))`
`thereforeS=1/2`
Hence, `S^(-1)=2`.