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Math Help - Find the sum of the series and determine the radius and interval of convergence

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    Find 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?
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    Hello,
    Quote Originally Posted by twilightstr View Post
    Let f(x)= ln (1+x^2). Find f^15(0)
    I know the answer is 0 but why?
    We know that \ln(1+x)=-\sum_{n\geq 1}\frac{x^n}{n}

    So here, 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 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
    \frac{f^{(15)}(0)}{15!}=0

    And hence f^{(15)}(0)=0
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