let-f-x-e-3x-x-2-4-developp-f-at-integr-serie- Tinku Tara June 4, 2023 Integration 0 Comments FacebookTweetPin Question Number 39214 by math khazana by abdo last updated on 03/Jul/18 letf(x)=e−3xx2+4developpfatintegrserie. Commented by math khazana by abdo last updated on 04/Jul/18 f(x)=∑n=0∞f(n)(0)n!xnletfindf(n)(0)f(x)=e−3x(x−2i)(x+2i)=e−3x4i{1x−2i−1x+2i}=14i{e−3xx−2i−e−3xx+2i}⇒f(n)(x)=14i{(e−3xx−2i)(n)−(e−3xx+3i)(n)}butleibnizformulagive(e−3xx−2i)(n)=∑k=0nCnk(1x−2i)(k)(e−3x)n−k)=∑k=0nCnk(−1)kk!(x−2i)k+1(−3)n−ke−3xalso(e−3xx+2i)(n)=∑k=0nCnk(−1)kk!(x+2i)k+1(−3)n−ke−3xf(n)(x)=14i∑k=0n(−1)kk!(−3)n−kCnk{1(x−2i)k+1−1(x+2i)k+1}f(n)(0)=14i∑k=0n(−1)kk!(−3)n−kCnn{1(−2i)k+1−1(2i)k+1}but1(−2i)k+1+1(2i)k+1=(2i)k+1−(−2i)k+14k+1=2iIm((2i)k+1)4k+1=2i2k+1ei(k+1)π22k+12k+1=i2kei(k+1)π2⇒f(n)(0)=14∑k=0n(−1)kk!(−3)n−kCnk12kei(k+1)π2andf(x)=∑n=0∞f(n)(0)n!xn Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: lim-x-1-2-3x-1-2-x-1-x-2-Next Next post: Question-170284 Leave a Reply Cancel replyYour email address will not be published.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.