How do you find a power series representation for f(x)=ln(1+x) and what is the radius of convergence?

1 Answer
Jun 27, 2018

ln(1+x) = sum_(n=0)^oo (-1)^n x^(n+1)/(n+1) for x in (-1,1)

Explanation:

Start from the geometric series:

sum_(n=0)^oo q^n = 1/(1-q)

converging for abs q < 1.

Let q=-t, then:

sum_(n=0)^oo (-t)^n = sum_(n=0)^oo (-1)^nt^n = 1/(1+t)

converging for abs t < 1

Inside the interval of convergence, that is for t in (-1,1), we can integrate term by term:

int_0^x dt/(1+t) = sum_(n=0)^oo (-1)^n int_0^x t^ndt

ln(1+x) = sum_(n=0)^oo (-1)^n x^(n+1)/(n+1)

obtaining a series with at least the same radius of convergence R=1.