1. ## taylor series help.

question:

Use integration in a term by term manner on this series for f(x)=1/(1+x) to find the taylor series for ln(1+x) about x=0

any help would be much appreciated thanks.

2. Originally Posted by Jonny-123
question:

Use integration in a term by term manner on this series for f(x)=1/(1+x) to find the taylor series for ln(1+x) about x=0

any help would be much appreciated thanks.
Well lets start by using there hint

$f(x)=\ln(1+x)$

Note that $f(0)=ln(1)=0$ we will need this later.

Now lets take the derivative to get

$f'(x)=\frac{1}{1+x}$

This is a geometric series with $r=(-1)$ so we get

$f'(x)=\sum_{n=0}^{\infty}(-1)^nx^n$

Now we integrate to get

$f(x)=C+\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}$

Since $f(0)=0 \implies C=0$ so we have

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}$