3.2 Introduction to Infinite Series


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1 3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are = 2, = 3 4, = 7 8, = 5 6,... These ifiite sequeces defied by sums are called ifiite series. Review of sigma otatio The Greek letter Σ used i this otatio idicates that we are addig ( summig ) elemets of a certai patter. (We used this otatio back i Calculus, whe we first looked at itegrals.) Here our sums may be ifiite ; whe this occurs, we are really lookig at a limit. Resources A itroductio to sequeces a stadard part of sigle variable calculus. It is covered i every calculus textbook. For example, oe might look at * sectio.3 (Itegral test),.4, (Compariso tests),.5 (Ratio & Root tests),.6 (Alteratig, abs. cov & cod. cov) i Calculus, Early Trascedetals (th ed., 2006) by Thomas, Weir, Hass, Giordao (Pearso) * sectio.3 (Itegral test),.4, (compariso tests),.5 (alteratig series),.6, (Absolute cov, ratio ad root),.7 (summary) i Calculus, Early Trascedetals (6th ed., 2008) by Stewart (Cegage) * sectios 8.3 (Itegral), 8.4 (Compariso), 8.5 (alteratig), 8.6, Absolute cov, ratio ad root, i Calculus, Early Trascedetals (st ed., 20) by Ta (Cegage) Itegral tests, compariso tests, ratio & root tests. * sectio 9.4 (Covergece Tests), 9.5 (Compariso, ratio, root tests), 9.6 (Alteratig, abs. cov & cod. cov) Calculus, Early Trascedetals (th ed., 2009) by Ato, Bives, Davis (Joh Wiley & Sos) p. 645 of Ato has a ice list. * sectio 0.3 (itegral test), 0.4 (alt series), 0.5 (compariso), 0.6 (absolute covergece), 0.7 (ratio ad root) i the Whitma College olie textbook: multivariable/ * Whitma s olie textbook: 0_Sequeces_ad_Series.pdf
2 3.2. What is a series? Give a ifiite series a we defie the partial sum S m := the partial sums = S = 2 a. Thus, i the series = = we have 2 S 2 = 3 4 We mea, by the expressio, S 3 = 7 8 S 4 = 5 6, the limit, as, of the partial sums S. = I this case, the partial sums appear to have the patter S = 2. So really meas lim m = Sice, i this case, the limit is, we say that A easy divergece test = 2 2 = lim m 2 m =. = 2 =. Ituitively, if a series is to coverge to a fiite limit L we would expect that evetually the terms we are addig up are cotributig very little to the series that at some poit the sum is close to L ad each ew term is ot chagig that. This argumet ca be made precise (but we wo t do that here.) This gives us a theorem, the th term divergece test : Theorem. (th term test) If a series a coverges the the limit, as goes to ifiity, of the terms a, must be zero. Although stated i terms of covergece, the theorem is really a statemet about divergece, for it is equivalet to the statemet: The th term divergece test: If i a series a the terms a do ot go to zero the the series does ot coverge! (This statemet is the cotrapositive of the statemet i the theorem.) Geometric series The complexity of our ivestigatio ito series (iitiated log ago by the Beroulli brothers ad Euler) ca be displayed by examiig the family of geometric series cetral to our uderstadig of series ad two iterestig sporadic series, the harmoic series ad the alteratig harmoic series. 2
3 I a earlier sectio, we examied the series 2 = = We cocluded, just by observatio, that the partial sums had the form 2 ad thus the series coverged to. Note that this series has this property: each term added o is exactly 2 of the previous term. A series which has a ratio r such that each ew term is exactly r times the previous term, is said to be geometric. (See for a geeral discussio of these series, icludig moder applicatios.) The mai idea. There is a ice way to work out a formula for the partial sum of a geometric series. I geeral, a geometric series has form a + ar + ar 2 + ar ar +... = ar where a is the first term ad r is the commo ratio betwee terms. Let us write ad so Notice that (Notice how most terms cacel!) Therefore S m := a + ar + ar 2 + ar ar m = rs m := ar + ar 2 + ar ar m + ar m = rs m S m := ar m a. S m = a(rm ) r = a( rm ). r =0 m =0 ar. ar If r > the the expressio a( rm ) does ot coverge ad so the geometric series does ot r coverge. If r = the the expressio a( rm ) r is udefied but it is easy to check that the partial sums are S m = ma ad so the series diverges to ifiity. If r = the the a( rm ) r does ot coverge ad so the geometric series does ot coverge. The partial sums alterate betwee a ad 0. But if r < the the a( rm ) r coverges to a r. Therefore the geometric series coverges to a r. This is importat eough to emphasize as a theorem. = Theorem. (Geometric series) If r < the But if r the a + ar + ar 2 + ar ar +... = ar diverges. =0 3 ar = a r. =0
4 The geometric series are cetral to the study of ifiite series. We will see later i this course that if a series is ot geometric, we will attempt (i a certai way) to preted it is geometric ayway! Sometimes this pretese gives us very useful iformatio. (This will motivate the ratio ad root tests.) The Harmoic Series The th term divergece test says that the terms of a series must go to zero if there is ay hope of the series covergig. Warig! It is temptig to believe i the coverse statemet. Is it true that if the terms go to zero the the series coverges? That would be ice but it is ot true i geeral. Here is a example a classical oe where the terms go to zero but the series diverge. The series is called the harmoic series. The harmoic series = (See for Wikipedia s discussio of this series, icludig a explaatio for its ame.) The terms go to zero as goes to ifiity. Yet this series diverges! We give oe argumet that this series diverges. This argumet is a compariso test. Aother argumet will be give later. Compare the series = ad the series ( ) + ( ) + ( ) +... I the secod series, ay expressio of the form, 2s < 2 s+, we replace by 2 s+. Sice 2 s+, we have a series which is smaller tha the harmoic series. Each partial sum of the secod series o bigger tha the partial sum of the harmoic series. But otice that, sice i the secod series there are 2 s terms of the form the we ca collect 2s+ them together to form the term 2. So the secod series becomes ( ) + ( ) + ( ) }{{} 8 terms +... = ( 2 ) + ( 2 ) + ( 2 ) ( 2 ) +... We cotiue, forever, to add 2 i the sum. So this secod series diverges (by the th term test, if you will.) But sice the harmoic series is forever larger tha this divergig series, the the harmoic series diverges! The harmoic series forms a ice couterpoit to the th term test for divergece. If the th term of a series goes to zero, the series still might ot coverge. The harmoic series is a ice example of this pheomeo. 4
5 3.2.5 The Alteratig Harmoic Series Oe more iterestig series: Cosider the alteratig harmoic series ( ) + = ( ) Note the effect of the expressio ( ) + ; it forces the sigs to alterate, so that we are addig positive, the egative terms. It turs out that a alteratig series, where the th term does to zero, does coverge! The alteratig harmoic series coverges ideed, we will see later that its sum is l Telescopig series There is aother type of geometric series we ru ito from time to time, where a little trick will give ot oly covergece, but the exact value of the limit of the series. Cosider, for example, the series ( + ) = We ca use the algebraic cocept of partial fractios (remember that?!) to rewrite So ( + ) = +. ( + ) = + = ( 2 ) + ( 2 3 ) + ( 3 4 ) Notice how the terms begi to cacel! We ca rewrite the sum as = + ( ) + ( ) + ( ) I this case, the series collapses ( telescopes ) ad oly the first term survives. So ( + ) =. 5
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