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Question Number 111766 by mathmax by abdo last updated on 04/Sep/20
calculate lim_(n→+∞) Σ_(k=1) ^n ln((n/(n+k)))^(1/n)
calculatelimn→+∞∑k=1nln(nn+k)1n
Answered by Dwaipayan Shikari last updated on 04/Sep/20
lim_(n→∞) (1/n)Σ^n log((1/(1+(k/n))))=∫_0 ^1 log((1/(1+x)))=−[xlog(1+x)]_0 ^1 +∫_0 ^1 (x/(1+x)) =−log(2)+1−[log(1+x)]_0 ^1 =1−2log(2)
limn→∞1n∑nlog(11+kn)=∫01log(11+x)=−[xlog(1+x)]01+∫01x1+x=−log(2)+1−[log(1+x)]01=1−2log(2)
Answered by mathmax by abdo last updated on 07/Sep/20
A_n =(1/n)Σ_(k=1) ^n ln((n/(n+k))) =(1/n) Σ_(k=1) ^n ln((1/(1+(k/n)))) ⇒A_n is a Rieman sum lim_(n→+∞) A_n =∫_0 ^1 ln((1/(1+x)))dx =−∫_0 ^1 ln(1+x)dx =_(1+x=t) =−∫_1 ^2 ln(t)dt =−[tlnt−t]_1 ^2 =−{2ln2−2+1} =−{2ln2−1} =1−2ln(2)
An=1n∑k=1nln(nn+k)=1n∑k=1nln(11+kn)⇒AnisaRiemansumlimn→+∞An=∫01ln(11+x)dx=−∫01ln(1+x)dx=1+x=t=−∫12ln(t)dt=−[tlnt−t]12=−{2ln2−2+1}=−{2ln2−1}=1−2ln(2)

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