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Question Number 44317 by abdo.msup.com last updated on 26/Sep/18

let u_n =∫_0 ^∞   ((t−[t])/(t(t+n)))dt  find a equivalent of u_n  when n→+∞

letun=0t[t]t(t+n)dtfindaequivalentofunwhenn+

Answered by tanmay.chaudhury50@gmail.com last updated on 26/Sep/18

((t−r)/(t(t+n)))=(1/(t+n))−(r/(t(t+n)))             =(1/(t+n))−(r/n)×(((t+n)−t)/(t(t+n)))               =(1/(t+n))−(r/n)×((1/t)−(1/(t+n)))                =(1/(t+n))+(r/n)×(1/(t+n))−(r/n)×(1/t)                =(1+(r/n))((1/(t+n)))−(r/n)×(1/t)  ∫_0 ^1 {(1+(0/n))((1/(t+n)))−(0/n)×(1/t)}dt+  ∫_1 ^2 (1+(1/n))((1/(t+n)))−(1/n)×(1/t)dt+...  ∫_r ^(r+1) (1+(r/n))((1/(t+n)))−(r/n)×(1/t)+...  =(1+(0/n))∣ln(t+n)∣_0 ^1 −(0/n)×∣lnt∣_0 ^1 +       (1+(1/n))∣ln(t+n)∣_1 ^2 −(1/n)×∣lnt∣_1 ^2 +...        (1+(r/n))∣ln(t+n)∣_r ^(r+1) +(r/n)×∣lnt∣_r ^(r+1) +...  =(1+(0/n)){ln(((1+n)/(0+n)))}−(0/n)×{ln1−ln0}+       (1+(1/n)){ln(((2+n)/(1+n)))}−(1/n)×{ln2−ln1}+..       (1+(r/n)){ln(((r+1+n)/(r+n)))}−(r/n){ln(r+1)−lnr}+..  =Σ_(r=0) ^∞ (1+(r/n)){ln(((r+1+n)/(r+n)))}−(r/n){ln(((r+1)/r))}

trt(t+n)=1t+nrt(t+n)=1t+nrn×(t+n)tt(t+n)=1t+nrn×(1t1t+n)=1t+n+rn×1t+nrn×1t=(1+rn)(1t+n)rn×1t01{(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)010n×lnt01+(1+1n)ln(t+n)121n×lnt12+...(1+rn)ln(t+n)rr+1+rn×lntrr+1+...=(1+0n){ln(1+n0+n)}0n×{ln1ln0}+(1+1n){ln(2+n1+n)}1n×{ln2ln1}+..(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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