Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 3. Geometric series

Transcription

1 Roberto s Notes o Ifiite Series Chapter 1: Sequeces ad series Sectio Geometric series What you eed to kow already: What a ifiite series is. The divergece test. What you ca le here: Everythig there is to kow about geometric series! The goal of this sectio may look very ambitious, but we shall achieve it! Geometric series e guably the most commo ad most importat type of series, oe which will be used repeatedly both i theory ad practice. Their ame comes from the sequece of the same ame. Defiitio A series is said to be geometric if it is geerated by a geometric sequece, that is, if it ca be writte i the form: 1 where a ad r e costat real umbers. Notice that if r 0, or if a 0, the series eds up beig a ifiite sum of 0 s: ot very iterestig! If we exclude these borig optios, we ca write a geometric series i a differet way, which turs out to be better at describig the key properties of geometric series. Defiitio A series is said to be geometric if it ca be writte i the form: 1 a 0 1 where a ad r e o-zero costat real umbers. Although I am callig this the operatioal defiitio, it is the defiitio that everyoe uses, ad so shall we. I just wated to justify ad clify the differece i sttig value betwee a geometric sequece ad a geometric series. With this i place, we ca state everythig we eed to kow about the covergece of a geometric series, as promised, ad i oe short fact! Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page 1

2 Proof If 1 Techical fact If r 1, the geometric series ad its limit is: 0 1 If r 1, the geometric series r, the sequece of terms is 0 a r 0 is coverget is diverget., which cosists of umbers r, stay costat at whose absolute values keep icreasig, or, if 1 a 0. This meas that such sequece does ot coverge to 0, ad therefore, the series diverges by the divergece test. If r 1, we use some algebra o the ptial sums. We have: k Sk a k 1 1 a r r r r 1 r If we expad the product o the umerator, we otice that all terms cacel, except for the first ad last, so that: S k a 1 r 1 r k1 Sice r 1, higher powers of r become smaller ad smaller. I the limit we have: as claimed. Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page k 1 a 1 r a lim Sk lim k k 1 r 1 r Ad that s it folks! All that is left is applyig this fact wheever we have a geometric series. Example: 5 0 We ca write this series as series with 1, thus recogizig it as a geometric a, r. Sice r 1, this series is coverget to: 5 a 15 1 r Example: 4.1 At first sight, this does ot look like a series, but it is! Beig a umber with a repeatig decimal expasio, we ca write it as: We ca make oe more chage that will reveal the ature of this umber as the sum of a series:

3 Et voila: a geometric series with a, r. It coverges ad we kow what its limit is, so we ca coclude that: You may recogize i the coclusio of this example the results of the procedure used to write a repeatig decimal as a ratioal umber. Ad here is aother applicatio of geometric series. Example: A ball is dropped from a iitial height of h metres above groud ad each time it hits the groud, it bouces to 90% of its previous height. How much distace will the ball travel i total? At first sight it may look like the ball will keep boucig forever, at least theoretically, ad that if our experiece tells us that the ball will evetually stop, it must be because this model is iaccurate or some other factor will make the ball stop. I fact, eve theoretically, the ball will oly cover a fiite total distace ad will evetually stop i fiite time! To see that, let us figure out how much distace the ball covers altogether. Whe we let it fall, it travels a distace of b0 h. Durig the first bouce it will go up by 0.9h ad the dow by the same b h. amout, for a total of same amout, for a total of b 0.9 h At the ext bouce it will go up by h ad the dow by the. Now we ca see the patter ad realize that the total distace is give by: b h h h But what we have here is a geometric series with 0.9. It is therefore coverget ad we kow what it coverges to. So we kow that the total distace will be: 0.9 h1 0.9 h1 19h How log will it take? All we have to do is figure out, i a simil way, how log it takes for each bouce to be completed ad add them all up. We shall get aother series, but will it be geometric? Fid out! We shall see ad use geometric series a lot more, but we already kow everythig about them. All you eed to do is practice eough to make them good frieds of yours. Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page

4 Summy A geometric series is obtaied by addig the terms of a geometric sequece. A geometric series coverges if ad oly if its commo ratio is less tha 1 i absolute value. a I that case, its limit is 1 r. Commo errors to avoid Geometric series e very simple ad very ice, but ot all series e geometric! Apply properties of geometric series to geometric series oly! Leig questios for Sectio S 1- Review questios: 1. Explai which series ca be called geometric.. Describe all the covergece properties of geometric series. Memory questios: 1. What is the geeral form of a geometric series?. What is the usual formula for the limit of a coverget geometric series?. Whe is a geometric series coverget? Computatio questios: Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page 4

5 For each of the series i questios 1-8: explai why it is geometric determie if it is coverget or diverget if it is coverget, compute its sum cos 5 t Of the two series ad cos oe coverges ad oe 1 si / 1 diverges. Determie which oe does what ad provide a valid gumet for your coclusio i each case. 11. Determie the values of k for which the series 1. Write the umber 5. as a fractio. 0 e k is coverget. 10. I your ext test, would you rather have a percetage mk of? 1 78 Justify your aswer! 40 or 1 1. Write the umber.1 as a fractio. 14. If the series a b is such that a 1 1 ad b e geometric sequeces, 1 ad its first three terms e 5/, 5/ ad 10/9, what is the geeral formula that describes a? Is the series coverget? Theory questios: Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page 5

6 1. Why do we study geometric series, but ot ithmetic series?. I the stadd formula for the geometric series, what does a represet? Proof questios: 1. Use geometric series to determie a geeral procedure to chage a repeatig decimal to a fractio. Applicatio questios: 1. Determie the total distace travelled by a ball dropped from a height of 0 feet, if the ball bouces to 95% of its previous height after each bouce.. Which series represets the time it takes for the ball of the previous questio to come to a complete stop? Is this series geometric? If so, compute its sum.. A collector offers a item for sale for $1000. A iterested buyer couteroffers $700. The seller proposes to split the differece, thus proposig $850. The buyer does the same ad offers $775. If they pla to cotiue i this way for ever, what price will they evetually agree o? 4. A used c salesma wats to sell a c for $1000. A repeat customer is iterested i that c ad the salesma kows that this customer will offer $00 less tha the iitial offer ad the they will play the game of split the differece util covergig o a fial price. What should the salesma s iitial request be? Templated questios: 1. Costruct a geometric series ad determie its covergece. If it coverges, determie its sum. What questios do you have for your istructor? Ifiite Series Chapter 1: Sequeces ad series Sectio : Geometric series Page 6