Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite su foud log the wy is clled prtil su. The su s bove of the first ters is clled the th prtil su. If we tke the su to ifiity the we hve ifiite series. But wht does it e to dd ifiite sequece of ubers d c such ifiite series dd up to fiite uber? I other words does ifiite series hve liit does it coverge? Defiitio: Give sequece { } d series if li ( ) exists the ( ) li d the series coverges. If the liit goes to or does ot exist the the series diverges. How c we deterie whether or ot series coverges? Oe wy to pproch this is to reeber tht we hve two sequeces ivolved i every series. The first is the sequece of ters { }. The secod is the sequece of prtil sus: s s s. If we c show tht the sequece of prtil sus { } Exple A: Does the series Ituitive Aswer: coverges to wht vlue? s coverges we will be ble to coclude tht the series coverges. coverge d if so to wht liit?
Be creful tht you re cler i your ow id bout the differeces og the sequece the sequece of prtil sus d the series. sequece { } 6 sequece of prtil sus { } s 6 series 6 Exple A exteded: Fid d. Aswers: coverges to ; coverges to Exple A exteded gi: Fid d. Aswers: coverges to ; to
Exple B: Deterie whether or ot the series 7 Aswer: coverges to coverges d if so fid its liit. Exple C: Deterie whether ( ) coverges d if so fid its liit. Aswer: diverges
The text proves couple of theores tht forlize the reltioship betwee series d its relted sequece. Theore 9. sttes: If coverges the li. The cotrpositive which is logiclly equivlet is stted s Corollry 9.9: If li or does ot exist the diverges. Note tht i Exples A d B bove (coverget series) li d li while i 7 Exple C (diverget series) ( ) tht li diverges sice li. does ot exist. Also fro Lecture 9. Exple A i we ow kow Iportt cutiory ote (gi): Be sure tht you re cler i your ow id tht there is differece betwee the sequece of ters for { }. d the sequece of prtil sus { } s produced by the series The coverse of Theore 9. is ot true: li is ot gurtee tht will coverge. Although the hroic sequece coverges to the hroic series diverges (see text! Exple ). Thus (Lecture 9. Exple B) sice the sequece coverges to lthough we c sy tht the series! ight coverge we cot be certi tht it does coverge. The text lso proves soe very coveiet theores which ese the tsk of evlutig series. Theore 9. sttes: For c d the geoetric series c clculte the su of the series: c r c r. r c r coverges if d oly if r < d if it coverges we The proof relies o clcultig the sequece of prtil sus d idetifyig the ptter which eerges. Exple A is geoetric series with c d ( ) d ( ) r d.
Theore 9. d ssocited theores give us es of cobiig series i uch the se wy tht we c dd (d subtrct) d ultiply (d divide) liits:. If d b both coverge the ( ) b coverges d ( ) b b. b. For y uber c if coverges the c lso coverges d c c. Exple D: Evlute. Aswer: coverges to Method: Use fctorig d seprtio of frctios to rerrge the series ito the for of geoetric series d use the forul fro Theore 9. to evlute: r c r c r with r <. ( ) ( ) fctor: seprte the frctios (distribute the divisio): rewrite s c r : pply Theore 9.: siplify: