Why can I use number 1 in Taylor series arctan?

Question

Why can I use number 1 in Taylor series arctan? Taylor serie arctan: $\sum_{n=1}^{\infty}(-1)^n \frac{x^{2n+1}}{2n+1}$

Answer 1

Yes, you can, because:

• By the alternating series test, that series converges when $x=1$.
• By Abel's theorem,$$\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}=\lim_{x\to1^-}\sum_{n=0}^\infty\frac{(-1)^nx^n}{2n+1}.$$