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Question Number 40889 by abdo.msup.com last updated on 28/Jul/18
prove?that ∫_0 ^1 ((1−(1−t)^n )/t)dt =Σ_(k=1) ^n (1/k)
prove?that∫011−(1−t)ntdt=∑k=1n1k
Answered by math khazana by abdo last updated on 30/Jul/18
we have 1−x^n =(1−x)(1+x+x^2 +...+x^(n−1) )⇒ ((1−(1−t)^n )/t) =((t(1+(1−t)+(1−t)^2 +...+(1−t)^(n−1) )/t) ∫_0 ^1 ((1−(1−t))/t)dt =∫_0 ^1 Σ_(k=0) ^(n−1) (1−t)^k dt =Σ_(k=0) ^(n−1) ∫_0 ^1 (1−t)^k dt =Σ_(k=0) ^n [−(1/(k+1))(1−t)^(k+1) ]_0 ^1 =Σ_(k=0) ^(n−1) (1/(k+1)) =Σ_(k=1) ^n (1/k) (=H_n ).
wehave1−xn=(1−x)(1+x+x2+…+xn−1)⇒1−(1−t)nt=t(1+(1−t)+(1−t)2+…+(1−t)n−1t∫011−(1−t)tdt=∫01∑k=0n−1(1−t)kdt=∑k=0n−1∫01(1−t)kdt=∑k=0n[−1k+1(1−t)k+1]01=∑k=0n−11k+1=∑k=1n1k(=Hn).

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