By using the power series for f(x)=(1)/(1-x), find the power series for g(x)=(2)/((1-x^)3). Options:

Question

By using the power series for
`f(x)=(1)/(1-x)`, find the power series for
`g(x)=(2)/((1-x^)3)`.

Options:

Answer 1

∑₂°° n (n − 1) xⁿ⁻²

Explanation:

f(x) = 1 / (1 − x)

f'(x) = 1 / (1 − x)²

f''(x) = 2 / (1 − x)³

Therefore:

g(x) = f''(x)

g(x) = d²/dx² [1 / (1 − x)]

Using sum of a geometric series:

g(x) = d²/dx² ∑₀°° xⁿ

g(x) = d/dx ∑₁°° n xⁿ⁻¹

g(x) = ∑₂°° n (n − 1) xⁿ⁻²