→ exboy11 :perfect! 09/04 09:59
※ 引述《marlboro001 (小學)》之銘言：
: 證明
: C(n,0) - (1/2)C(n,1) + (1/3)C(n,2) - .... +{[(-1)^n]/(n+1)}C(n,n) = 1/(n+1)
: 其中
: C(n,i) = n!/i!(n-i)!
n
Σ (-1)^k * C(n,k)/(k+1) =1/(n+1)
k=0
pf:
n
Σ (-1)^k * C(n,k)/(k+1)
k=0
n C(n,k) n n!
= Σ (-1)^k * ----------- = Σ (-1)^k * ----------------------
k=0 (k+1) k=0 (k+1)* k!* (n-k)!
n n!
= Σ (-1)^k * ----------------
k=0 (k+1)! *(n-k)!
n n! *(n+1)
= Σ (-1)^k * ---------------------------
k=0 (k+1)! *(n-k)! *(n+1)
1 n (n+1)!
= ---------* Σ (-1)^k * ----------------
(n+1) k=0 (k+1)! *(n-k)!
1 n
= ---------* Σ (-1)^k *C(n+1,k+1)
(n+1) k=0
-1 n
= ---------* Σ (-1)^(k+1) *C(n+1,k+1)
(n+1) k=0
-1 n
= ---------* ( { Σ (-1)^(k+1) *C(n+1,k+1)} -1 )
(n+1) k+1=0
-1 1
= ---------* ( 0 -1 ) = ----------
(n+1) (n+1)
since:
n
0=(1-1)^n= Σ (-1)^k * C(n,k)
k=0
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※ 編輯: h2o1125 來自: 218.187.61.204 (07/05 17:55)