Let $F(x)=\displaystyle \int^x_0\frac{1}{1+t^3}dt$
1)Prove that F is well defined and differentiable for all $x\in\mathbb{R}$
2)If $n$ is a positive integer show that $F(x)=\displaystyle(\sum\limits_{k=0}^n (-1)^k\frac{x^{3k+1}}{3k+1})+(-1)^{n+1}\int_0^x\frac{t^{3n+3}}{1+t^3}dt$
3)Prove that $F(x)=\displaystyle\sum\limits_{k=0}^\infty (-1)^k\frac{x^{3k+1}}{3k+1}$ for all $|x|\le1$
For part one I'm assuming by well defined it means the integral exists which is direct since the function under integration is continuous, but the differentiability part is not clear I'm not sure how I should approach the limit. As for parts 2,3 I've never encountered any similar problems( I think I should take the limit as $n\to\infty$ in 3)