Lecture 09.pdf - 3 7.2 test Comparison integrals Improper fat flxldx = fab fins fcxidx = it f \u21d2 E > o IN b Ice \u21d2 t O I Companion cc d Def Proof I b

# Lecture 09.pdf - 3 7.2 test Comparison integrals Improper...

• 11

This preview shows page 1 - 4 out of 11 pages.

3. 7.2 Comparison test Improper integrals fat " flxldx = fins , fab fcxidx = figs , ,IEb ) Icb ) = fabflxidx it converges if the limit exist f E > o , IN such that b , c > N I Icb ) Icc , I =/ f ! fix ) dx ) a E supine ¥yaftx ' =D fab fcxidx = final ! fcxtdx-lciga.IE ) Ice ) = f ! fan dx it converges it the limit exists t O , It 870 . a cc , d cats I Icc ) Icd ) I = I fcdfcxidxl CE Them I Companion test ) If I full sgcxl for all X C- Ca , b ) and fabglxldx converges fabfcxidx also converges ( actually it is absolutely convergent ) Def fabfcxidx is called absolutely convergent if fabffcxydx converges Proof : I the case b = tool f ? gcxldx converges fat ? fcxidx converges b > a , Icb ) = fab fix ) dx b. c > a I Icb ) - Icc ) I =/ f ! fix ) dx ) S f ! lflxildx s fbcgcxidx Since fat 's glxldx is convergent , t 270 , 7. N , c > b > N I I gcxldx C E 11lb ) - Ital CE
Ex If ! ° 19¥ dx ( P > o ) if p > I , say p =L -128 870 - compare 17¥ with ¥+5 P 8=1+871 I EI × Lit . '7÷¥ - fig . . xs time . = tests o s so 7- lb ? o.o , 13,7¥ a I tx > b × P S i. e , 19¥ a ftp.sfjoxptsdx converges Sgt 's toff dx converges