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

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- May 7th 2008, 10:53 AMkithyTaylor ans Maclaurin series
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 - May 7th 2008, 11:32 AMflyingsquirrel
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.

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

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

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

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

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

$\displaystyle \ldots$

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