Question

Find the sum to n terms of the series 1/2+3/2^2+5/2^3+...+(2n-1)/2^n?

Answer 1

See below.

Explanation:

`sum_(k=0)^n (2k+1)x^k = 2xsum_(k=0)^n kx^(k-1)+sum_(k=0)^nx^k`

but

`sum_(k=0)^n x^k = (x^(n+1)-1)/(x-1)` and

`sum_(k=0)^n kx^(k-1) = d/(dx)sum_(k=0)^n x^k = (1 + (n (x-1)-1) x^n)/(x-1)^2` then

`sum_(k=0)^n (2k+1)x^k = 2x((1 + (n (x-1)-1) x^n)/(x-1)^2)+ (x^(n+1)-1)/(x-1)`

and finally making `x = 1/2`

`sum_(k=0)^n (2k+1)2^-k=6 - 2^-n (5 + 2 n)`