数学技巧(一)

数学技巧(一)

级数

  • n = 1 n 2 x n = x n = 1 ( n + 1 ) n x n 1 x n = 1 n x n 1 = x ( n = 1 x n + 1 ) x ( n = 1 x n ) \sum \limits_{n=1}^{\infty}n^2x^n=x\sum \limits_{n=1}^{\infty}(n+1)nx^{n-1}-x\sum \limits_{n=1}^{\infty}nx^{n-1}=x(\sum \limits_{n=1}^{\infty}x^{n+1})''-x(\sum \limits_{n=1}^{\infty}x^n)'

线性代数

  • ( A E ) 1 = B E A E = ( B E ) 1 A = ( B E ) 1 + E = ( B E ) 1 ( E + B E ) (A-E)^{-1}=B-E \Rightarrow A-E = (B-E)^{-1} \Rightarrow A = (B-E)^{-1}+E = (B-E)^{-1}(E+B-E)
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