Taylor ans Maclaurin series

**kithy** · May 7th 2008, 10:53 AM

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

**flyingsquirrel** · May 7th 2008, 11:32 AM

Hi

The Taylor series expanded about 1 is $\sum_{n\geq 0}\frac{f^{(n)}(1)}{n!}(x-1)^n$ ... but we don't know $f^{(n)}(1)$ . I suggest you try to guess what is $f^{(n)}(x)$ using the first derivatives and then you'll show it using induction.

$f^0(x)=f(x)=\ln(1+x)$

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

$f^2(x)=-\frac{1}{(1+x)^2}$

$f^3(x)=\frac{1}{(1+x)^3}$

$f^4(x)=-\frac{1}{(1+x)^4}$

$\ldots$

$f^n(x)= \,?$

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