By integrating the series for 11+x2\frac{1}{1+x^2}1+x21, identify the series for arctanx\arctan xarctanx.
∑n=0∞(−1)nx2n+12n+1\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}n=0∑∞2n+1(−1)nx2n+1
∑n=0∞x2n+12n+1\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n+1}}{2n+1}n=0∑∞2n+1x2n+1
∑n=0∞(−1)nx2n2n\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{2n}n=0∑∞2n(−1)nx2n
∑n=1∞(−1)nx2n2n+1\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n x^{2n}}{2n+1}n=1∑∞2n+1(−1)nx2n