[安乐椅#9] 常见裂项
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根式裂项
\(\dfrac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}\)
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分数裂项
\(\dfrac{1}{n(n+1)}=\dfrac{1}{n}-\dfrac{1}{n+1}\)
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三元分数裂项
\(\dfrac{1}{n(n+1)(n+2)}=\dfrac{1}{2}[\dfrac{1}{n(n+1)}-\dfrac{1}{(n+1)(n+2)}]\)
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倒数平方裂项
\(\dfrac{2n+1}{n^2(n+1)^2}=\dfrac{1}{n^2}-\dfrac{1}{(n+1)^2}\)
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平方差裂项
\(2n-1=(n+1)^2-n^2\)
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立方差裂项
\(3n^2+3n+1=(n+1)^3-n^3\)
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对数裂项
\(\log_a \dfrac{n+1}{n}=\log_a(n+1)-\log_an\)
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上幂基本裂项
\(\dfrac{(n-1)2^n}{n(n+1)}=\dfrac{2^{n+1}}{n+1}-\dfrac{2^n}{n}\)
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上幂平方裂项
\(\dfrac{(n^2-2n-1)2^n}{n^2(n+1)^2}=\dfrac{2^{n+1}}{(n+1)^2}-\dfrac{2^n}{n^2}\)
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下幂基本裂项
\(\dfrac{n+2}{n(n+1)2^{n+1}}=\dfrac{1}{n2^n}-\dfrac{1}{(n+1)2^{n+1}}\)
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下幂平方裂项
\(\dfrac{n^2+4n+2}{n^2(n+1)^22^{n+1}}=\dfrac{1}{n^22^n}-\dfrac{1}{(n+1)^22^{n+1}}\)
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积裂项
\(n(n+1)=\dfrac{1}{3}[n(n+1)(n+2)-(n-1)n(n+1)]\)
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单比临偶裂项
\(\dfrac{n}{n^4+n^2+1}=\dfrac{1}{2}[\dfrac{1}{n(n-1)+1}-\dfrac{1}{n(n+1)+1}]\)
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单比阶乘裂项
\(\dfrac{n}{(n+1)!}=\dfrac{1}{n!}-\dfrac{1}{(n+1)!}\)
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基积幂裂项
\((n+1)2^n=(n+1)2^{n+1}-n2^n\)
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平方积幂裂项
\((n^2+4n+2)2^n=(n+1)^22^{n+1}-n^22^n\)
