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.
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
$\displaystyle f(x)=\ln(1+x)$
Note that $\displaystyle f(0)=ln(1)=0$ we will need this later.
Now lets take the derivative to get
$\displaystyle f'(x)=\frac{1}{1+x}$
This is a geometric series with $\displaystyle r=(-1)$ so we get
$\displaystyle f'(x)=\sum_{n=0}^{\infty}(-1)^nx^n$
Now we integrate to get
$\displaystyle f(x)=C+\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}$
Since $\displaystyle f(0)=0 \implies C=0$ so we have
$\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{(-1)^nx^{n+1}}{n+1}$