given f(x) = ln (1+x)
a) determine the taylor series for f(x) expanded about x=1
b) what is the interval of convergence for the series in part A
thanks a lot for your help
K
Hi
The Taylor series expanded about 1 is $\displaystyle \sum_{n\geq 0}\frac{f^{(n)}(1)}{n!}(x-1)^n$... but we don't know $\displaystyle f^{(n)}(1)$. I suggest you try to guess what is $\displaystyle f^{(n)}(x)$ using the first derivatives and then you'll show it using induction.
$\displaystyle f^0(x)=f(x)=\ln(1+x)$
$\displaystyle f'(x)=\frac{1}{1+x}$
$\displaystyle f^2(x)=-\frac{1}{(1+x)^2}$
$\displaystyle f^3(x)=\frac{1}{(1+x)^3}$
$\displaystyle f^4(x)=-\frac{1}{(1+x)^4}$
$\displaystyle \ldots$
$\displaystyle f^n(x)= \,?$