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

Explanation

For `x != 1`, let
`S = x + 3x^(2) + 5x^(3) + ... + (2n - 3) x^(n - 1) + (2n - 1) x^(n)`...(i)
`rArr xS = x^(2) + 3x^(3) + ...+ (2n - 5) x^(n - 1) + (2n - 3) x^(n) + (2n - 1) x^(n + 1)`...(ii)
Subtracting (ii) from (i), we get
`(1 - x) S = x + 2x^(2) + 2x^(3) + ... + 2x^(n - 1) + 2x^(n) - (2n - 1) x^(n + 1) = x + (2x^(2) (1 - x^(n - 1)))/(1 - x) - (2n - 1) x^(n + 1)`
`=(x)/(1 - x) [1 - x + 2x - 2x^(n) - (2n -1) x^(n) + (2n - 1) x^(n + 1)]`
`rArr S = (x)/((1 - x)^(2)) [(2n - 1) x^(n + 1) - (2n + 1) x^(n) + 1 + x]`
Thus `((2n + 1)/(2n - 1)) + 3((2n + 1)/(2n - 1))^(2) + ....+ (2n -1) ((2n + 1)/(2n -1))^(n)`
`= ((2n + 1)/(2n -1))((2n - 1)/(2))^(2) [(2n - 1) ((2n + 1)/(2n -1))^(n+ 1) - (2n + 1) ((2n + 1)/(2n -1))^(n) + 1 + (2n + 1)/(2n - 1)]`
`= (2n^(2) - 1)/(4) .(4n)/(2n -1) = n (2n + 1)`