[安乐椅#9] 常见裂项

1. 根式裂项

   $\dfrac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}$

2. 分数裂项

   $\dfrac{1}{n(n+1)}=\dfrac{1}{n}-\dfrac{1}{n+1}$

3. 三元分数裂项

   $\dfrac{1}{n(n+1)(n+2)}=\dfrac{1}{2}[\dfrac{1}{n(n+1)}-\dfrac{1}{(n+1)(n+2)}]$

4. 倒数平方裂项

   $\dfrac{2n+1}{n^2(n+1)^2}=\dfrac{1}{n^2}-\dfrac{1}{(n+1)^2}$

5. 平方差裂项

   $2n-1=(n+1)^2-n^2$

6. 立方差裂项

   $3n^2+3n+1=(n+1)^3-n^3$

7. 对数裂项

   $\log_a \dfrac{n+1}{n}=\log_a(n+1)-\log_an$

8. 上幂基本裂项

   $\dfrac{(n-1)2^n}{n(n+1)}=\dfrac{2^{n+1}}{n+1}-\dfrac{2^n}{n}$

9. 上幂平方裂项

   $\dfrac{(n^2-2n-1)2^n}{n^2(n+1)^2}=\dfrac{2^{n+1}}{(n+1)^2}-\dfrac{2^n}{n^2}$

10. 下幂基本裂项

    $\dfrac{n+2}{n(n+1)2^{n+1}}=\dfrac{1}{n2^n}-\dfrac{1}{(n+1)2^{n+1}}$

11. 下幂平方裂项

    $\dfrac{n^2+4n+2}{n^2(n+1)^22^{n+1}}=\dfrac{1}{n^22^n}-\dfrac{1}{(n+1)^22^{n+1}}$

12. 积裂项

    $n(n+1)=\dfrac{1}{3}[n(n+1)(n+2)-(n-1)n(n+1)]$

13. 单比临偶裂项

    $\dfrac{n}{n^4+n^2+1}=\dfrac{1}{2}[\dfrac{1}{n(n-1)+1}-\dfrac{1}{n(n+1)+1}]$

14. 单比阶乘裂项

    $\dfrac{n}{(n+1)!}=\dfrac{1}{n!}-\dfrac{1}{(n+1)!}$

15. 基积幂裂项

    $(n+1)2^n=(n+1)2^{n+1}-n2^n$

16. 平方积幂裂项

    $(n^2+4n+2)2^n=(n+1)^22^{n+1}-n^22^n$