Roberto s Notes on Infinite Series Chapter 1: Series Section 2. Infinite series

Transcription

1 Roberto s Notes o Ifiite Series Chapter : Series Sectio Ifiite series What you eed to ow already: What sequeces are. Basic termiology ad otatio for sequeces. What you ca lear here: What a ifiite series is. Some basic related termiology ad otatio. The topic of ifiite series is extremely iterestig ad rich i its ow right, but we shall develop it oly to a small extet ad i relatio to calculus. Our mai iterest is i the problem of how to itegrate (idefiitely) those fuctios whose atiderivative caot be writte as a fiite combiatio of elemetary fuctios, such as y x e. The questio we shall try to aswer is: sice we caot write these atiderivatives as fiite combiatios of elemetary fuctios, ca we write them as a ifiite combiatio? Maybe, but first we eed to clarify what we mea by a ifiite combiatio, or more specifically a ifiite sum, ad for that we eed, you guessed it, limits ad sequeces. Our startig poit is a sequece a, whose terms we wat to add. But how ca we add ifiitely may umbers? Eve though additio is commutative, it turs out, as we shall see soo, that whe we try to add up ifiitely may umbers, what we get may deped o the order i which we add them. Ad of course we have the problem of the time it ca tae to add such a large amout of umbers! So, we start from the small ad familiar. Give a sequece by the fiite sum: Defiitio a S, its -th partial sum is give a Therefore, each sequece a geerates a sequece of partial sums: S a Notice that i this defiitio the order i which the terms of the sequece appear plays a importat role. We are ot just addig the terms i ay which way, but exactly i the order i which they appear i the sequece. This will become a importat aspect of the theory of series. Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page

2 I particular ulie what happes for fiite sums, addig up ifiitely may umbers is ot always a commutative operatio. We are ow ready to defie a series. Give a sequece a Defiitio, its associated ifiite series, or just its series, is the expressio of the form a, defied as the limit of its sequece of partial sums: a S lim S lim a If such limit exists we say that the series is coverget to S. If ot, we say that the series is diverget. The sequece o which a series is based is called its sequece of terms. Example: This series is geerated by the sequece the partial sums: S 4 8,,, 4 8, through Does it coverge? To figure this out, otice that we ca write each partial sum as: S Sice fiite sums are associative, we ca write this expressio as: 4 4 All the terms i bracets cacel, leavig oly: S As we tae the limit of this partial sum as goes to ifiity, we get. Therefore the series coverges to. This example uses a method that we shall geeralize i a later sectio o telescopig series. So, the covergece of a ifiite series is equivalet to the covergece of its sequece of partial sums. Sice the covergece of a sequece is ot easy to chec i geeral, it may loo that determiig if a series is coverget may be eve more difficult. I fact it is t, sice the fact that a series is defied through a sum provides additioal tools that ca be used effectively. However, the first ad easiest criterio we ca use to aalyze covergece relies o checig the covergece of the origial sequece. If a series Techical fact a is coverget, the the sequece a must coverge to 0. Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page

3 Proof If the series is coverget, its sequece of partial sums must be such. This meas that the partial sums must become closer ad closer to the limit L. But this meas that as we go from oe partial sum to the ext, the step we tae must become smaller ad smaller, evetually becomig 0. But such steps are exactly the terms of the origial sequece, which must therefore coverge to 0. By cosiderig the opposite of this fact, we obtai the first test for the covergece of a series, a divergece test, i fact, that should always be used first wheever we aalyze a series. Techical fact The divergece test If the sequece a is ot coverget to 0, the the series Example: a is diverget. 3 The terms that defie this series approach, which meas that i the series we eep addig terms that are closer ad closer to. Hece the partial sums icrease by almost at every step ad caot possibly coverge to a fiite value. oly oe directio! A commo misuse of it cosists of applyig it, icorrectly, bacwards. Kot o your figer The divergece test provides a implicatio i oly oe directio ad therefore ca oly lead to a coclusio of divergece. The fact that a sequece a is coverget to 0 tells us othig about the covergece of the series a. The typical example used to illustrate the above warig is based o a very importat series, oe that will be used repeatedly to examie ad illustrate properties of series. The series defied by: Defiitio S 3 4 is called the harmoic series. We shall see deeper ad more subtle tests later, but this oe will still prove to be a very powerful tool. But BE CAREFUL: this is a divergece test ad wors i Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 3

4 Proof Techical fact The sequece of terms of the harmoic series coverges to 0, but the series itself is diverget. We ow that lim 0 it requires the sequece NOT to coverge to 0., so that the divergece test caot be used, sice But this does ot mea that the series is coverget. To see that it is ot, cosider a large partial sum: S We ca rewrite it as: S We ca ow observe that: S But sice the series icludes ifiitely may terms, we ca obtai as may of these s as we wat, which meas that by taig a large eough, we ca mae the partial sum as large as we wat. Therefore the series diverges to. So, if we are dealig with a series whose sequece of terms coverges to 0, we may be dealig with a coverget series, as i we have see with the harmoic series. We simply do ot ow., or with a diverget oe, as Before looig at other importat types of series ad other covergece or divergece tests, tae some type to play with some simple series, so as to clarify the cocept, use the divergece test ad mae sure you do t use it bacwards, thus ruig the ris to get to icorrect coclusios. Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 4

5 Summary A series is a ifiite sum, defied as the limit of a sequece of partial sums. Sice a series is the limit of a sequece, the order i which its terms are added may be importat. I order for a series to coverge, its sequece of terms must coverge to 0. Commo errors to avoid Be clear o the fact that the value of a series is actually a limit, subject to all we ow about limits. The divergece test is a oe-directioal test: do ot use it to claim covergece, as it is ot capable of doig that. Learig questios for Sectio S - Review questios:. Describe what a series is. 3. Describe what the harmoic series is ad why it does ot coverge.. Explai the relatioships betwee a series ad its sequece of terms. Memory questios:. What is a series?. What is the th partial sum of a series? 3. Whe is a series coverget? 5. State the divergece test. 6. Which series is called harmoic? 7. Is the harmoic series coverget? 4. Do the properties of a series deped o the order of its terms? Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 5

6 Computatio questios: For each of the series i questios -0: a) idetify the first four terms of its sequece of terms b) idetify the first four terms of its sequece of partial sums c) use the divergece test to see if the series is diverget a l 3. cos A certai series S a is such that S. Determie the formula that describes the geeric term of the sequece a as a sigle, proper fractio ad decide whether the series coverges.. A sequece a is defied by 0 values of the first 5 partial sums of the series a. a a 3, a. Determie the exact A sequece is defie recursively by a, a a. Use sigma 3 otatio to describe the series whose terms are give by this sequece. Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 6

7 Theory questios:. What is the mai questio of iterest whe studyig ifiite series?. Which sequece must coverge i order for a series to coverge? 3. If lim a 0, does it follow that a is coverget? 4. Accordig to the divergece test, what ca we say about the sequece of terms of a series that is diverget? 5. If the series a follow that the series cotais oly positive terms ad is coverget, does it a is diverget? 7. Is it possible for a series to diverge if its partial sums are bouded? 8. If two series a ad b are both diverget, does it imply that a b is diverget too? 9. Kowig the startig value of i a series is ot always importat. For what purpose is it actually eeded? 0. Is it true that the partial sums of a series are its terms whe viewed as a sequece?. Why is it acceptable for a series to start at 3, istead of startig at? 6. If we ow the partial sums S of a series, how do we compute each of its terms? (Do t leave ay cases out!) Proof questios:. Show that if 0 the series 0 e is diverget. Templated questios:. Aalyze ay series you ca fid or costructed i the same way you did for Computatio questios -0. Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 7

8 What questios do you have for your istructor? Ifiite Series Chapter : Sequeces ad series Sectio : Ifiite series Page 8