What is a power series representation for $f(x)=ln(1+x)$ and what is its radius of convergence?

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Answer 1 · 2018-03-29T13:14:13

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

with radius of convergence $R=1$ .

Start from the sum of the geometric series:

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

converging for $abs q < 1$ .

Let $x = -q$ to have:

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

Inside the interval of convergence $x in (-1,1)$ we can integrate the series term by term:

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

and obtain a series with the same radius of convergence $R=1$ :

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

What is a power series representation for f(x)=ln(1+x)f(x)=ln(1+x) and what is its radius of convergence?