How to find $f(x)$ in order to find $f^{(10)}(3)$?

Question

(a) Find the radius of convergence of $\sum_{n=1}^\infty (-1)^n \frac{(x-3)^n}{(2n+1)}$ and its derivative.

(b) Denote by $f(x)$ the function represented by the above power series within its region of convergence. Find $f^{(10)}(3)$, i.e., its 10th derivative at $x = 3$.

I can solve problem (a). However I cannot find f(x). In order to find f(x) how should I do?

Answer 1

You don't need to find $f(x)$. To do part (b), you only need to think about the coefficients of the Taylor series of $f(x)$ at $x=3$.

Answer 2

Hint: If we have $$ f(x) = \sum_{n \ge 0} a_n (x-a)^n, $$ then the $n$th coefficient $a_n$ is also equal to $\frac{f^{(n)}(a)}{n!}$.

Answer 3

Hint: $$\frac{\arctan \sqrt{x}}{\sqrt{x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{n}\quad {\text{ for }}|x|\leq 1,x\not =\pm i\!$$ $$x\rightarrow (x-3)$$ $$\frac{\arctan \sqrt{x-3}}{\sqrt{x-3}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}(x-3)^{n}\quad {\text{ for }}|x-3|\leq 1$$