Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li hold dow the Ctrl ey ad clic.) Defiitios Defiitio A series is a sequece of terms that you ited to add up. A fiite series has a fiite umber of terms ad the sum is well-defied ad idepedet of the order i which the terms are added: a a a a a A ifiite series has a ifiite umber of terms ad hece the sum is ot ecessarily well-defied ad may i fact deped o the order i which the terms are added. Whether or ot the sum is well-defied we still write the series as: a a a a The iitial idex may or may ot be. If we do t say whether a series is fiite or ifiite, we ormally mea a ifiite series. We will later give a precise way to add a ifiite series, but we first give a example of the problems that ca arise if you add a ifiite series icorrectly: Example What is wrog with the followig proof that 0? 0 0 0 0 0 0 0 Solutio: The associative rule does ot wor for a ifiite sum. Remar: Thus if we add the terms of the sequece a we get a expressio of the form a a a a which is called a series ad is deoted by a. Defiitio The Sum of the Series Defiitio Give a ifiite series S a, its th -partial sum is the fiite series
S a. The, the sum of the ifiite series is defied to be S lims or a lim provided the limit exists or is positive or egative ifiity. a Remar This says that a ifiite sum must be computed i the order the terms are listed i the series: a a a a a 4 Remar Thus, a ifiite series is associated with two sequeces: the sequece of terms: a ad the sequece of partial sums: S. The sum of the series (or simply the series) is the sum of the sequece of terms ad is the limit of the sequece of partial sums. Further Termiology: If the limit exists (i.e. S lims is fiite), we say the series exists or is coverget or coverges to S or has sum S. If the limit does ot exist (i.e. lims does ot exist), we say the series does ot exist or is diverget or diverges or does ot have a sum. If the limit is positive ifiity (i.e. lims ), we say the series diverges to. If the limit is egative ifiity (i.e. lims ), we say the series diverges to. To say that the limit is positive or egative ifiity does ot say that the limit exists! It merely says the way i which it does ot exist, i.e. the way i which it diverges. Geometric Series Fiite Defiitio costat. A geometric series is a series i which the ratio of successive terms is a We begi with fiite geometric series: Example The fiite series S 4 6 64 is geometric ad the ratio of successive terms is. The sumis S 9. This series may be writte i summatio otatio i may ways such as:
6 S 5 0 7 5 I ay case, the first term is 4, the ratio of successive terms is, there are 6 terms ad the sum is 9. 0 4 We write the geeral fiite geometric series as S ar a ar ar ar ar 0 although may other forms are possible. I this series, the first term is a, the ratio of successive terms is r ad there are terms. Summig the Series Gauss foud a way to write dow the geeral sum without usig a summatio ( ) or a ellipsis (). Proceed as follows: Multiply the series by r: rs ar ar ar ar ar Subtract the formula for rs from the formula for S ad otice that all terms cacel except the first term i S ad the last term i rs : S rs a ar If r, this may be solved for S: S a r r If r, the origial series may be easily summed: S a a a a a 0 sice there are terms each of which is a. I summary, the geeral fiite geometric series is S ar a ar ar ar ar 0 a r r if r a if r where the first term is a, the ratio of successive terms is r ad the umber of terms is.
6 Example For our origial example S, fid the sum by usig the geeral formula. Solutio: The first term is 4, the ratio of successive terms is ad there are 6 terms. So the sum is 4 6 6 S 4 64 9. Notice that we do ot eed to write the summatio with the idex startig at 0 before idetifyig the first term, the ratio, or the umber of terms. Example 4 4. 4 is a Geometric Series where a 6 ad r 4. Its sum 6 4 Your tur: Exercise 7 Compute p p...... Geometric Series Applicatios Exercise A ball is dropped from a height of feet. Each time it bouces it reaches a height which is of the height o the previous bouce.. What is the total distace travelled by the ball (o the ifiite umber of bouces)?.......... What is the total time the ball taes to travel this distace?....... Exercise The spiral at the right is made from a ifiite umber of semicircles whose ceters are all o the x-axis. The first semicircle is cetered at x ad has radius r. The radius of each subsequet semicircle is half of the radius of the previous semicircle. 0.5.5. Cosider the ifiite sequece of poits where the spiral crosses the x-axis. What is the x-coordiate of the limit of this sequece?.......... What is the total legth of the spiral (with a ifiite umber of 4
semicircles)? Or, is the legth ifiite?......... Telescopig Series Telescopig series are aother class of series which ca be summed exactly. Ufortuately, there is o precise defiitio of a telescopig series. Suffice it to say that: part of each term cacels with part of oe or more subsequet terms allowig oe to explicitly compute the partial sums ad hece the total sum of the series. The best way to uderstad telescopig series is through examples. Example Compute S. Solutio: To see what is happeig, we first write out the first six terms i two ways. O the oe had, we combie the fractios: 6 0 0 4 I this form it is very hard to tell what the sum is. However, i the origial form we have 4 4 5 Part of each term cacels part of the ext term. However, we caot use the associative rule. So we eed to loo at the partial sums. We compute the th partial sum ad cacel everythig except the first half of the first term ad the last half of the last term: S So the sum of the series is S Covergece ad Divergece Tests 4 lim S lim Why is it importat to ow whe a series coverges? Because, for example: Coverget Series May be Used to Defie ad Approximate Fudametal Costats Mathematicias ofte use series to compute decimal values for fudametal costats (lie ad e)or to defie ew fudametal costats. Here are some examples. Examples of Coverget Series Used to Defie ad Approximate Fudametal Costats 5
You are ot yet expected to be able to prove the covergece of these series or to estimate the error i the approximatio. Example Compute! 6 4 0 0 Remar Recall that factorial is! ad by defiitio 0!. Solutio: This series is either geometric or telescopig. So you do ot yet ow how to compute the sum. However, it ca be show to coverge (by the Ratio Test). Taig terms the partial sum is 0 S 0 0!.74590455 which ca be show to be correct to withi 09 (usig the Taylor Boud o the Remaider). I fact, Taylor s theorem shows that the sum of the ifiite series is e. So this is a excellet way to fid a decimal approximatio to e. Example Compute 7 64 5 Solutio: This series is called the p-series with p. It ca be show to coverge (by the Itegral 5 Test). Taig 5 terms the partial sum is S 5.09 which ca be show to be correct to withi 0 (usig the Itegral Boud o the Remaider). Riema recogized that the sum of this ifiite series is a ew trascedetal umber (ot expressible i terms of ad e) ad amed it, which is read zeta of. Example Cosider the series S 5 4 4 where later terms are each half of the precedig term. Is this series coverget or diverget ad why? Solutio: If we igore the first three terms, the tail is the geometric series,,, 4,, whose ratio is. Cosequetly, the tail coverges ad so the series coverges. I fact the sum of the geometric tail is 4. So the sum of the origial series is S 5 4 4 6. Defiitio A series a is positive if all of its terms are positive, a 0 for all. o A series a is egative if all of its terms are egative, a 0 for all. o A series a is idefiite if some terms are positive ad some terms are egative. o With these examples i mid, we ca ow tur to the Covergece ad Divergece Tests. There is oe test which ca show that a series is diverget. There are tests which ca be used for series whose 6
terms are all positive. Ad there are tests which ca be used for series whose terms are both positive ad egative. I additio, there is oe geeral priciple, used all the time, which we discuss ow. It says that to determie the covergece of a series, we ca igore ay umber of iitial terms ad oly loo at the remaiig part of the series, called the tail. Defiitio TestigaTail A tail of the series i a is ay series of the form a where N i. N A series a is coverget if ad oly if ay (ad hece every) tail is coverget. o Remar This says the covergece of a series does ot deped o ay fiite umber of terms. Further, you ca chec for covergece by applyig a covergece test to the tail. Example Cosider the series S 5 4 4 where later terms are each half of the precedig term. Is this series coverget or diverget ad why? Solutio: If we igore the first three terms, the tail is the geometric series,,, 4,, whose ratio is. Cosequetly, the tail coverges ad so the series coverges. I fact the sum of the geometric tail is 4. So the sum of the origial series is S 5 4 4 6. th Term Divergece Test Propositio The th Term Divergece Test If lim a 0, the a is diverget. o Remar If lim a 0the th Term Divergece Test FAILS ad says othig about a ; it may be coverget or diverget. Example Cosider the series Sice lim, this series diverges. Example The Harmoic Series. has lim 0 as the limit, but the series diverges. (We shall show this soo.) o 7
Example diverges sice The series lim 0 Example The series 4 9 6 5 diverges because lim a does ot exist. The Itegral Test The Itegral Test: Suppose f is a cotiuous, positive, decreasig fuctio o, ad let a f. The the series a is coverget if ad oly if the improper itegral fxdx is coverget. For the proof hold dow the Ctrl ey ad clic o Itegral Test Example The Harmoic Series. Cosider fx x o,; thefx is positive ad decreasig ad x dx lim x dx lim l x lim l l Sice this itegral diverges, the give series diverges by the Itegral Test. Example The p-series p coverges if p. Cosider fx x p o,; thefx is positive ad decreasig ad x p dx lim x p dx lim x p p lim p p p which diverges if p sice i this case p 0 so that we are taig the limit of to a positive power as that power goes to. The series coverges if p, sice i this case we are taig the limit of to a egative power as that power goes to. [The case whe p reduces the p-series to the Harmoic Series]. Example Show that the series
l coverges. Solutio: Note that the series begis at, sice l 0. 0, so let fx l xl x. The xl x l Thus the series coverges by the Itegral Test. Itegral Test Remaider Estimate Theorem. The Itegral Test Estimate. Suppose that a is a series which satisfies the hypotheses of the Itegral Test usig the fuctio f ad which coverges to L. Let be the th partial sum ad let The s a a a r L s fxdx r fxdx For the proof ad examples of this theorem hold dow the Ctrl ey ad clic o Itegral Test Estimate 9