Power Series (Not a test question)

Question

By using the power series for  , find the power series for  .

Answer 1

Notice that f'(x) = g(x)

So, find the power series for f(x) and then differentiate it term by term.

1/ (1+x) = 1/(1-(-x)) is the sum of the geometric series ∑_{(n=0 to infinity)}(-x)ⁿ =

1 - x + x² - x³ + ...

So, g(x) = f'(x) = -1 + 2x - 3x² + ... = ∑_{(n=1 to infinity)}(-1)ⁿnx^n-1