series question?

Question

Let S denote the infinite sum 2 + 5x + 9x^2 + 14x^3 + 20x^4 + ...., where |x| < 1 and the coefficient of x^(n-1) is ½ n (n + 3), (n = 1, 2, ....). Then S equals ?

can anybody tell,how to solve this kind of problem ?

my book has given a procedure to do s-x*s

why shall i do that ? does the problem tells any ideas to go about ?

someone please explain

Answer 1

S = sum from n = 1 to infinte of 1/2n(n + 3)x^(n -1)

or S = sum from n = 0 to inf of 1/2(n + 1)(n + 4)x^n

then xS = sum from 1 to inf of 1/2n(n + 3)x^n

and
S = 2 + sum from 1 to inf of 1/2(n + 1)(n + 4)x^n

then S - xS = 2 + sum from 1 to inf of :
1/2 [(n + 1)(n + 4) - n(n + 3)]x^n = (n + 2)x^n

S = 2 + sum from 1 to inf (n + 2)x^n

= sum from 0 to inf of (n + 2)x^n

= sum from 0 to inf of nx^n
+ sum from 0 to inf of 2x^n

= x*sum from 0 to inf of nx^(n - 1)
+ 2 sum from 0 to inf of x^n

sum from 0 to inf of x^n = 1 / (1 - x)

sum from 0 to inf of nx^(n - 1) = sum from 0 to inf of (x^n)' (derivate) then equal to
[1 / (1 - x)] ' = 1 / (1 - x)²

then S = x / (1 - x)² + 2 /(1 - x)
= (2 - x) / (1 - x)²