I was exploring the Wikipedia page on Geometric series wherein they've used the formula for sum to compute the power series for $\tan^{-1}x$. Geometric Power Series -Wikipedia
$$\int \dfrac{1}{1+x^2}\mathrm dx=\int\sum_{k=0}^{\infty}(-1)^kx^{2k}\mathrm dx=\sum_{k=0}^{\infty}\dfrac{(-1)^k}{2k+1}x^{2k+1}$$
The computation of the integral of the sum makes perfect sense to me but isn't the formula for sum namely $S_n=\displaystyle\sum_{i=0}^{\infty}ar^i=\left(\dfrac{a}{1-r}\right)$ valid when $\mid r\mid \lt1$.
Why do we simply neglect the fact that the sum converges iff $\mid r\mid \lt1$? Or does it mean that the Power Series for $\tan^{-1} x$ would converge for $\mid x\mid \lt1$? Thanks in anticipation.