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Question Number 165725 by mathocean1 last updated on 06/Feb/22

Show that : ∀ k ∈ N^(∗ ) , (1/(k+1))≤ln(k+1)−ln(k)≤(1/k)

Showthat:∀k∈N∗,1k+1⩽ln(k+1)−ln(k)⩽1k

Answered by TheSupreme last updated on 07/Feb/22

ln(1+x)≤x ∀R^+ f(x)=ln(1+x) g(x)=x f(0)=g(0)=0 f′(x)=(1/(1+x)) g′(x)=1 f′(x)≤1=g(x) so f(x) grows lower than g(x) no other zeroes ln(k+1)−ln(k)=ln(((k+1)/k))=ln(1+(1/k))≤(1/k) −(ln(k)−ln(k+1))=−ln((k/(k+1)))=−ln(1−(1/(k+1)))≤−(1/(k+1)) ln(1−(1/(k+1)))≥(1/(k+1)) (1/(k+1))≤ln(k+1)−ln(k)≤(1/k)

ln(1+x)⩽x∀R+f(x)=ln(1+x)g(x)=xf(0)=g(0)=0f′(x)=11+xg′(x)=1f′(x)⩽1=g(x)sof(x)growslowerthang(x)nootherzeroesln(k+1)−ln(k)=ln(k+1k)=ln(1+1k)⩽1k−(ln(k)−ln(k+1))=−ln(kk+1)=−ln(1−1k+1)⩽−1k+1ln(1−1k+1)⩾1k+11k+1⩽ln(k+1)−ln(k)⩽1k

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