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Question Number 44317 by abdo.msup.com last updated on 26/Sep/18
letun=∫0∞t−[t]t(t+n)dtfindaequivalentofunwhenn→+∞
Answered by tanmay.chaudhury50@gmail.com last updated on 26/Sep/18
t−rt(t+n)=1t+n−rt(t+n)=1t+n−rn×(t+n)−tt(t+n)=1t+n−rn×(1t−1t+n)=1t+n+rn×1t+n−rn×1t=(1+rn)(1t+n)−rn×1t∫01{(1+0n)(1t+n)−0n×1t}dt+∫12(1+1n)(1t+n)−1n×1tdt+...∫rr+1(1+rn)(1t+n)−rn×1t+...=(1+0n)∣ln(t+n)∣01−0n×∣lnt∣01+(1+1n)∣ln(t+n)∣12−1n×∣lnt∣12+...(1+rn)∣ln(t+n)∣rr+1+rn×∣lnt∣rr+1+...=(1+0n){ln(1+n0+n)}−0n×{ln1−ln0}+(1+1n){ln(2+n1+n)}−1n×{ln2−ln1}+..(1+rn){ln(r+1+nr+n)}−rn{ln(r+1)−lnr}+..=∑∞r=0(1+rn){ln(r+1+nr+n)}−rn{ln(r+1r)}
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