20/03/26 15:42:53.55 IM17g/m8.net
もうちょっと頑張ってみた.
Σ[k=1,n] C{n,k}(-1)^{k+1}/k
= Σ[k=1,n] ∫[x=0,1]dx C{n,k} (-x)^{k-1}
= Σ[k=1,n] ∫[x=0,1]dx C{n,k} (x-1)^{k-1}
= ∫[x=0,1]dx { (1+(x-1))^n - 1 } / (x-1)
= ∫[x=0,1]dx (x^n - 1) / (x-1)
= ∫[x=0,1]dx Σ[k=0,n-1] x^k
= Σ[k=1,n] 1/k
Σ[j=1..n]a[j]/j
= Σ[j=1..n] C{n,j} ∫[x=0,1]dx ∫[y=0,1]dy (-xy)^{j-1}
= ∫[x=0,1]dx ∫[y=0,1]dy {1 - (1-xy)^n }/{1 - (1-xy)}
= ∫[x=0,1]dx Σ[k=0,n-1] ∫[y=0,1]dy (1-xy)^k
= ∫[x=0,1]dx Σ[k=1,n] (1-(1-x)^k) / kx
= ∫[x=0,1]dx Σ[k=1,n] (1-x^k) / k(1-x)
= Σ[k=1,n] 1/k ∫[x=0,1]dx Σ[m=0,k-1] x^m
= Σ[k=1,n] 1/k Σ[m=1,k] 1/m
= Σ[k=1,n]Σ[m=1,k] 1/km
εやらlog展開を使うなんてアリエネーだろ... と自分でも思っていたのでスッキリした.