Question

If `|x|gt1`,then sum of the series `(1)/(1+x)+(2)/(1+x^(2))+(2^(2))/(1+x^(4))+(2^(3))/(1+x^(8))+"......"" upto n terms "oo` is `(1)/(x-lambda)`,then the value of `lambda` is

Answer 1

Let `P=lim_(n to oo)((1)/(1+x)+(2)/(1+x^(2))+(2^(2))/(1+x^(4))+""....." upto n terms ")`
`=lim_(n to oo)sum_(r=0)^(n)((2^(r ))/(1+x^(2^(r )))+(2^(r ))/(1-x^(2^(r )))-(2^(r ))/(1-x^(2^(r ))))`
`=lim_(n to oo)sum_(r=0)^(n)((2^(r+1))/(1-x^(2^(r+1)))-(2^(r ))/(1+x^(2^( r))))`
`=lim_(n to oo)((2^(n+1))/(1-x^(2^(n+1)))-(1)/(1+x))`
`=lim_(n to oo)((2^(n+1))/(1-x^(2^(n+1))))/((1)/(x^(2^(n+1))-1))-(1)/(1+x)=0-(1)/(1-x)`
`=(1)/(x-1)=(1)/(x-lambda) " " [" given "]`
`:.lambda=1`.