8.3 Sequences & Series: Convergence & Divergence

Transcription

1 8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of the fuctio re clled the terms i the sequece,, such tht,, 3,,, Where is clled the th term (or rule of sequece), d we deote the sequece by. The sequece c be expressed by either ) mple umber of terms i the sequece, seprted by comms ) explicit fuctio defied by the rule of sequece 3) the rule of sequece set off i brces. Exmple : The sequece,4,6,8, is the sequece of eve umbers. Express the sme sequece s rule of oegtive iteger. The sequece,3,5, is the sequece of odd umbers. Express the sme sequece s rule of o-egtive iteger. How my i the list re eeded to estblish the rule i the bsece of the explicitly-stted rule? ***NOTE: Whe give sequece s list, the first term is usully desigted to be ssocited with. This is becuse we re usig s ordil (or coutig) umber, rther th crdil umber. We will be primrily iterested i wht hppes to the sequece for icresigly lrge vlues of. Exmple : 4 If 3, list out the first five terms, the estimte lim. Pge of 5

2 FACT: Let be sequece of rel umbers. Possibilities: ) If lim, the diverges to ifiity ) If lim, the diverges to egtive ifiity 3) If lim c, fiite rel umber, the coverges to c 4) If lim oscilltes betwee two fixed umbers, the diverges by oscilltio Defiitio:! is red s fctoril. It is defied recursively s Por ejemplo: 9! !! or s!! 3 3 Exmple 3: Determie whether the followig sequeces coverge or diverge. (),, 3, 4,,, (b),,,,,, (c) (d) (e) l (f)!! (g)!! (h) (i) (j)! (k) (l)! Pge of 5

3 Sometimes, lbeit rrely, we hve to write the rule of sequece s fuctio of from ptter. Exmple 4: Write expressio for the th term. () 3, 8, 3, 8,... (b) 5, -5, 45, -35,... (c), 4, 9, 6, 5,... (d) 4, 0, 8, 8,... (e), 3 3, 4 5, 5 7, 6,... (f) l, l, l4, l8, 9 A Series is the sum of the terms i sequece. Fiite sequeces d series hve defied first d lst terms, wheres ifiite sequeces d series cotiue idefiitely. A series is iformlly the result of ddig y umber of terms from sequece together: 3. A series c be writte more succictly by usig the summtio symbol sigm,., the Greek letter S for Esum (the E is both silet d ot relly there.) For ifiite series, we c look t the sequece of prtil sums, tht is, lookig to see wht the sums re doig s we dd dditiol terms. I geerl, the th prtil sum of series is deoted S. This c be explored o clcultor by ddig sequetil terms to the ggregte sum. Exmple 5: For both d b, geerte the sequece of prtil sums S, S, S3,, S, for ech, the determie if the sequeces coverges or diverges. Do the results surprise you? Where else hve we see somethig like this before? Pge 3 of 5

4 Covergece d Divergece of Series Wht does it me for series to coverge? To diverge? Let s look t couple series from specil fmily clled geometric series. Exmple 6: Give the series , fid the first te terms of the sequece of prtil sums, d list them below, S, S, S3,, S 0. Bsed o this sequece of prtil sums, do you thik the series coverges? Diverges? To wht? (HINT: first rewrite the rule of sequece so tht it looks like expoetil fuctio of.) Exmple 7: Give the series, fid the first five terms of the sequece of prtil sums, d list them below. Bsed o this sequece of prtil sums, do you thik the series coverges? Diverges? To wht? Pge 4 of 5

5 We re ow goig to look t severl fmilies of ifiite series d severl tests tht will help us determie whether they coverge or diverge. For some tht coverge, we might be ble to give the ctul sum, or itervl i which we kow the sum will be. For others, simply kowig tht they coverge will hve to suffice. Geometric Series, th Term Test for Divergece, d Telescopig Series Geometric Series Test (GST) A geometric series is i the form r or 0 The geometric series diverges if r. r, 0 If r, the series coverges to the sum S. r Where is the first term, regrdless of where strts, d r is the commo rtio. Exmple 8: Usig the GST, determie whether the followig series coverge or diverge. If the coverge, fid the sum. () 3 (b) 3 (c) 3 Exmple 9: Determie if the followig series coverge or diverge. If coverget, fid the sum. () 7 (b) 0.05 (c) (d) (e) Pge 5 of 5