直接裂项
$$\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$
$$\frac{1}{n(n+2)}=\frac{1}{2}(\frac{1}{n}-\frac{1}{n+2})$$(求和时隔一项消)
因式分解后裂项
$$\frac{1}{n^2+3n+2}=\frac{1}{(n+1)(n+2)}$$
分离常数后裂项
$$\frac{n^2+n+1}{n(n+1)}=1+\frac{1}{n(n+1)}$$
指数型裂项
$$\frac{2^n}{(2^n-1)(2^{n+1}-1)}=\frac{1}{2^n-1}-\frac{1}{2^{n+1}-1}$$
三项
$$\frac{1}{n(n+1)(n+2)}=\frac{n+1}{n(n+1)(n+1)(n+2)}=\frac{1}{2}(\frac{1}{n(n+1)}-\frac{1}{(n+1)(n+2)})$$
$$\frac{n+2}{n(n+1)2^n}=\frac{(n+2)2^{n+1}}{n2^n(n+1)2^{n+1}}$$
裂成加法
$$(-1)^n\frac{2n+1}{n(n+1)}=(-1)^n(\frac{1}{n}+\frac{1}{n+1})$$
特殊裂项
$$lg\frac{n+1}{n}=lg(n+1)-lgn$$
$$\frac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}$$
$$nn!=(n+1-1)n!=(n+1)!-n!$$