Let f(x)= ln (1+x^2). Find f^15(0)

I know the answer is 0 but why?

- May 14th 2009, 11:22 AMtwilightstrFind the sum of the series and determine the radius and interval of convergence
Let f(x)= ln (1+x^2). Find f^15(0)

I know the answer is 0 but why? - May 14th 2009, 11:35 AMMoo
Hello,

We know that $\displaystyle \ln(1+x)=-\sum_{n\geq 1}\frac{x^n}{n}$

So here, $\displaystyle f(x)=-\sum_{n\geq 1}\frac{x^{2n}}{n}$

So you can notice that the factors of the odd powers of x are 0.

But we also know that $\displaystyle f(x)=\sum_{n\geq 0} \frac{f^{(n)}(0)}{n!} \cdot x^n=f(0)+\sum_{n\geq 1} \frac{f^{(n)}(0)}{n!} \cdot x^n=\sum_{n\geq 1} \frac{f^{(n)}(0)}{n!} \cdot x^n$

So by identification of the 15-th term

$\displaystyle \frac{f^{(15)}(0)}{15!}=0$

And hence $\displaystyle f^{(15)}(0)=0$