\arctan(x+1)=\frac{\pi}{4}+\Big(\frac{x}{2\cdot 1}-\frac{x^2}{2\cdot 2}+\frac{x^3}{4\cdot 3}\Big)-\Big(\frac{x^5}{8\cdot 5}-\frac{x^6}{8\cdot 6}+\frac{x^7}{16\cdot 7}\Big)+\cdots=\frac{\pi}{4}+\sum_{n=0}^{\infty}\frac{(-1)^nx^{4n+1}}{2^{2n+2}}\Big(\frac{2}{4k+1}+\frac{2x}{4k+2}+\frac{x^2}{4k+3}\Big).