QuestionUse term-by-term differentiation or integration to find a power series centered at $x=0$ for: $$f(x)=\tan ^{-1}\left(x^{9}\right)=\sum_{n=0}^{\infty}$$ $\square$
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Studdy Solution
The resulting power series for $f(x) = \tan^{-1}(x^9)$ centered at $x = 0$ is:
$$f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{18n+9}}{2n+1}$$
This series represents the function $\tan^{-1}(x^9)$ for $|x^9| \leq 1$, which simplifies to $|x| \leq 1$.
The power series centered at $x = 0$ for $f(x) = \tan^{-1}(x^9)$ is:
$$\boxed{\sum_{n=0}^{\infty} (-1)^n \frac{x^{18n+9}}{2n+1}}$$