求 sigma 四次方及五次方的公式?

Σn^4 = (6n^5 + 15n^4 + 10n^3 - n)/30

Σn^5 = (2n^6 + 6n^5 + 5n^4 - n^2)/12

Σn^6 = n(n+1)(2n+1)(3n^4+6n^3-3n+1)/42

使用積分方程式推導公式: a,b = 待定係數 = ?

Σn^(k+1) = a*∫(Σn^k)*dn + b*n


Ex1: Σ1 = n

Σn = ?

= a∫n*dn + b*n

= a*n^2/2 + b*n


邊界條件:

1 = a/2 + b

3 = 2a + 2b

=> a = 1, b=1/2

=> Σn = n^2/2 + n/2 = n(n+1)/2



Ex2: Σn^2 = a∫(n^2+n)/2 + b*n

= a*(n^3/6 + n^2/4) + b*n

邊界條件:

1 = 5a/12 + b

5 = 7a/3 + 2b

=> a = 2, b = 1/6

=> Σn^2 = 2*(n^3/6 + n^2/4) + n/6 = n(n+1)(2n+1)/6


Ex3. Σn^3 = a∫(2n^3+3n^2+n)/2 + b*n

= a*(n^4/2 + n^3 + n^2/2) + b*n

邊界條件:

1 = 2a + b

9 = 18a + 2b


=> a = 1/2, b = 0

=> Σn^3 = (n^4/2 + n^3 + n^2/2)/2

= (n^2 + 2n + 1)*n^2/4

= [n(n+1)/2]^2


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