Question-62455 Tinku Tara June 4, 2023 Integration 0 Comments FacebookTweetPin Question Number 62455 by aliesam last updated on 21/Jun/19 Commented by mathmax by abdo last updated on 22/Jun/19 S=∑n=1∞AnwithAn=∫n−π2n+π2e−x∣cos(x)∣dxchangementx=n−π2+tgiveAn=∫0πe−(n−π2+t))∣cos(π2−(n+t))∣dt=eπ2−n∫0πe−t∣sin(n+t)∣dtifwesupposesin(n+t)⩾0on[0,π](thatneedaproof)⇒An=eπ2−n∫0π(sin(n)cos(t)+cos(n)sin(t))dt=sin(n)eπ2−n∫0πcostdt+cos(n)eπ2−n∫0πsintdt=0+cos(n)eπ2−n[−cost]0π=2cos(n)eπ2−n⇒S=2∑n=1∞cos(n)eπ2−n=2eπ2∑n=1∞e−ncos(n)=2eπ2(∑n=0∞e−ncos(n)−1)but∑n=0∞e−ncosn=Re(∑n=0∞e−n+in)=Re(∑n=0∞(e−1+i)n)wehave∣e−1+i∣=1e<1⇒∑n=0∞(e−1+i)n=11−e−1+i=11−e−1(cos(1)+isin(1))=11−e−1cos(1)−ie−1sin(1)=1−e−1cos(1)+e−1sin(1)(1−e−1cos(1))2+e−2sin2(1)⇒∑n=0∞e−ncos(n)=1−e−1cos(1)(1−e−1cos(1))2+e−2sin2(1)⇒S=2eπ2{1−e−1cos(1)1−2e−1cos(1)+e−2−1}. Commented by aliesam last updated on 22/Jun/19 thankyousirbrilliantsolution Commented by mathmax by abdo last updated on 22/Jun/19 youarewelcomesir. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Find-the-remainder-when-2014-is-divisible-by-2017-Next Next post: Question-62456 Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.