Representations of functions as power series

Question

Find a power series representation for the function.

f(x) = (8+x)/ (1-x)²

^{f(x) = ∑(from n=0 to ∞) ( ? )}

Answer 1

1 / (1-x) = sum of geometric series with a = 1 and r = x

= ∑_{(n=0 to ∞)} arⁿ = 1 + x + x² + x³ + ...

So, d/dx[1 /(1-x)] = 1/(1-x)² = 1 + 2x + 3x² + ...

Multiply by 8+x: (8+x)/(1-x)² = 8[1/(1-x)²] + x[1/(1-x)²]

= 8(1 + 2x + 3x²+ ...) + x(1 + 2x + 3x² + ...)

= 8 + 17x + 26x² + 35x³ + ... (note that 8, 17, 26, 35, ...is arithmetic)

So, (8+x) / (1-x)² = ∑_{(n=1 to ∞)}(9n-1)x^n-1, or equivalently,

∑_{(n=0 to infinity)} (9n+8)xⁿ.