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Question Number 31460 by abdo imad last updated on 08/Mar/18
findintermsofnthevalueofAn=∫01∏k=1n−1(x2−2xcos(kπn)+1)dxwithnfromN★.
Commented by abdo imad last updated on 10/Mar/18
letdecomposeinsideC[x]p(x)=x2n−1therootsofp(x)arezk=eikπnwithk∈[0,2n−1]]wehavez0=1,z1=eiπn,z2=ei2πn,...zn−1=ei(n−1)πn,zn=−1,zn+1=ei(n+1)πn=z−n−1,zn+2=z−n−2,z2n−1=z−1⇒p(x)=∏k=02n−1(x−zk)=(x2−1)∏k=1n−1(x−zk)(x−z−k)=(x2−1)∏k=1n−1(x2−2Re(zk)x+∣zk∣2)=(x2−1)∏k=1n−1(x2−2cos(kπn)x+1)⇒∏k=1n−1(x2−2cos(kπn)x+1)=x2n−1x2−1⇒An=∫01x2n−1x2−1dx=∫01(1+x2+x4+...+x2n−2)dxAn=[x+13x3+15x5+....+12n−1x2n−1]01=1+13+15+.....+12n−1letfindAnintermsofHnAn=1+12+13+15+...+12n−1+12n−12−14−...−12nAn=H2n−12HnwithHn=∑k=1n1k.
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