20/03/25 00:53:30 /JBpNTYO.net
>>899
Σ[k=1..n]a[k] = 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}
=lim[ε→∞] ∫[x=ε,1]dx ( x^{-1} - x^{-1}(1-x)^n )
=lim[ε→∞] ∫[x=ε,1]dx x^{-1} - ∫[x=0,1-ε]dx (1-x)^{-1} x^{n} )
=lim[ε→∞] -log(ε) - Σ[k=n+1..∞](1-ε)^k/k
=lim[ε→∞] -log(ε) - Σ[k=1..∞](1-ε)^k/k + Σ[k=1..n](～)
=lim[ε→∞] -log(ε) + log(1-(1-ε)) + Σ[k=1..n](～)
= Σ[k=1..n] 1/k
= {Σ[k=1..n] 1/k - ln(n)} + ln(n)
lim[n→∞]Σ[k=1..n]a[k]/ln(n) = 0.5772.../∞ + 1 = 1