Determine the maximal compact interval such that $\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$ holds true

Question

The Assignment:

Determine the maximal compact interval, such that the following identity holds true:$$\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1} = \arctan(x)$$ Explain your answer and show explicitly, why your result is maximal.

We are currently discussing function series and how uniform convergence stands in relation with swapping limits, derivatives, integrals (...). We've had a lot of results in the lectures recently, but we can't really see how to apply them here. Furthermore we actually already know this identity from last year for $|x| ≤ 1$, so we think that we're kind of missing the point in this assignment as of now.

Any help would be appreciated.

Answer 1

By the ratio test we see that the radius of convergence is $R=1$ and by the Leibniz's rule we have the convergence at $x=\pm1$ hence the interval of convergence is the compact $[-1,1]$ which's maximal since for all $x\not\in [-1,1]$ the series is divergent.