Which of the following is a power series expression for the given fraction ?

Question

Which of the following is a power series expression for  ?

	X2 + x3 + x4 + x5 + …
	x2 + 2x3 + 3x4 + 4x5 + …
	x + x3 + x5 + x7 + x9 + …
	x2 + x4 + x6 + x8 + …

Answer 1

x / (1 - x²) has the form of the sum of the geometric series with first term a = x and common ratio r = x².

So, the given expression is the sum of the series ∑_{(n=0 to infinity)} arⁿ = ∑_{(n=0 to infinity)} x(x²)ⁿ =

x + x³ + x⁵ + ...