証明(1)C(n,0)+(...=【1/(n+1)】*【2^

(1) Notice that

(1 + x)^n = Σ (nCk)x^k

Integrate both sides w.r.t x from 0 to 1

[2^(n + 1) - 1]/(n + 1) = Σ (nCk)/(k + 1)

(2) (1 + x)^n = Σ (nCk)x^k

Integrate both sides w.r.t x from -1 to 0

1/(n + 1) = Σ (nCk)(-1)^k / (k + 1)

Expressed in full form

C(n,0)-(1/2)*C(n,1)+(1/3)*C(n,2)-...+(-1)^n【1/(n+1)】*C(n,n)=【1/(n+1)】