常见公式及定理

\(\ln(x+1)=\sum_{n=1}^\infty\dfrac{(-1)^{n+1}}{n}x^n\)

\(\ln(1-x)=\sum_{n=1}^\infty\dfrac{-1}{n}x^n\)

\(\sum_{i=1}^ni^2=\dfrac{n\cdot(n+1)\cdot(2n+1)}{6}\)

\(\sum_{i=1}^ni^3=[\dfrac{n\cdot(n+1)}{2}]^2\)

\(\int_{0}^{1}{\left(x-x^{2}\right)^{n}}\mathrm{d}x=\sum_{i=0}^n(-1)^n*\dfrac{C_n^i}{n+i+1}=\dfrac{(n!)^2}{(2n+1)!}\)