Why can I use number 1 in Taylor series arctan?

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

• Your question is unclear. – Fred Jul 16 at 17:57
• Do you mean: why can $x$ be $1$? If so because that is within its radius of convergence. – badjohn Jul 16 at 18:00
• @badjohn It's on the boundary. Convergence is not guaranteed. – saulspatz Jul 16 at 18:14
• @saulspatz Good point. I guess that I should have said: it's known to converge for $1$. – badjohn Jul 16 at 18:25

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