Re: 請教一題馬克勞林級數

[mathmanliu (煩惱即是菩提)]

※ 引述《ADAH33 (逐漸消失的生命)》之銘言：
:  expand f(x)=(1+x)/(1-x)^3 in a Maclaurin series

公式: 1/(1-x)^n = Σ H(n, k) x^k    k=0,1,2,....,    -1<x<1

f(x)= (1+x) Σ H(3,k)x^k    , k=0,1,...

       ∞             ∞
    = Σ H(3,k)x^k  + Σ H(3,k)x^(k+1)
       0              0
         ∞             ∞
    = 1+ Σ H(3,k)x^k + Σ H(3,k) x^(k+1)
         1              0
         ∞                     ∞
    = 1+ Σ H(3, k+1) x^(k+1) + Σ H(3,k) x^(k+1)
         0                      0
          ∞  4+2k
    = 1 + Σ ─── H(3, k+1) x^(k+1)
          0   3+k

      ∞
    = Σ (k+1)^2 x^k
       0

Note: H(n,k) = C(n+k-1, k)

 4+2k            2(k+2) (k+3)(k+2)
───H(3,k+1) = ───────── = (k+2)^2
 3+k              (k+3)*1*2

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