Limits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3
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1 Limits of sequeces I this uit, we recall what is meat by a simple sequece, ad itroduce ifiite sequeces. We explai what it meas for two sequeces to be the same, ad what is meat by the -th term of a sequece. We also ivestigate the behaviour of ifiite sequeces, ad see that they might ted to plus or mius ifiity, or to a real limit, or behave i some other way, I order to master the techiques explaied here it is vital that you udertake plety of practice exercises so that they become secod ature. After readig this text, ad/or viewig the video tutorial o this topic, you should be able to: use the otatio for a sequece i terms of a formula for its -th term; decide whether a sequece teds to ifiity; decide whether a sequece teds to mius ifiity; decide whether a sequece teds to a real limit; decide whether a sequece diverges; use the otatio for the limit of a sequece. Cotets 1. Itroductio 2 2. Some otatio for sequeces 3 3. The behaviour of ifiite sequeces 3 1 c mathcetre July 18, 2005
2 1. Itroductio A simple sequece is a fiite list of umbers. For example, is a simple sequece. So is (1, 3, 5,..., 19) (4, 9, 16,..., 81). We call the umbers i the sequece the terms of the sequece. So i our secod example we say that the first term is 4, the secod term is 9, ad so o. Not all sequeces have to be fiite. We ca also defie ifiite sequeces. These are lists of umbers, like simple sequeces, but the differece is that the terms go o for ever. We write ifiite sequeces like this: (2, 5, 8,...). I a fiite sequece, the three dots i the middle followed by the fial umber just mea that we have omitted some of the terms. But if you see three dots without aythig after them, it meas that the sequece goes o for ever. We say two sequeces are the same if all the terms are the same. This meas that the sequeces have to cotai the same umbers, i the same places, throughout the sequece. So (1, 2, 3, 4,...) (2, 1, 4, 3,...) eve though they cotai the same umbers. The positios of the umbers are differet, so the sequeces are ot the same. Our first two examples of sequeces have obvious rules for obtaiig each term. I the first example, we obtai the -th term by takig, multiplyig by 2, ad takig away 1. We say that the -th term is 2 1. Similarly, the -th term of the secod sequece is ( + 1) 2. Our third example, the ifiite sequece, also has rules for the -th term. I this case we take, multiply by 3, ad take away 1. So the -th term is 3 1. But ot all sequeces have clear rules for the -th term. For example, the sequece that starts ( 3, 5, 99 7,...) is still a sequece, eve though it looks radom. Key Poit A simple sequece is a fiite list of umbers, ad a ifiite sequece is a ifiite list of umbers. The umbers i the sequece are called the terms of the sequece. Two sequeces are the same oly if they cotai the same umbers i the same positios. c mathcetre July 18,
3 2. Some otatio for sequeces You might have oticed that we have writte out our sequeces i brackets. For example, we have writte (1, 3, 5, 7,..., 19) for our first sequece. If a sequece is give by a rule the aother, more cocise, way to deote the sequece is to write dow the rule for the -th term, i brackets. So we might write this sequece as (2 1). But we also eed to show how may terms are icluded i the sequece, ad we do this by writig the idex of the first term just udereath the bracket, ad the idex of the last term just above, like this: (2 1) 10 =1. Similarly, we could write the fiite sequece (4, 9, 16, 25,..., 81) as (( + 1) 2 ) 8 =1. Notice that we use two sets of brackets for this sequece. The ier set is used to deote the formula ( + 1) 2 for the -th term. The outer set idicates that we have a sequece, icludig all the terms with this formula from = 1 to = 8. We ca use the same otatio for ifiite sequeces. Our example of a ifiite sequece ca be writte as (3 1) =1. Here, the superscript is used to idicate that the sequece goes o for ever. Key Poit We deote a sequece by writig the -th term i brackets ad idicatig how may terms are icluded i the sequece. 3. The behavior of ifiite sequeces We shall ow cocetrate o ifiite sequeces. It is ofte very importat to examie what happes to a sequece as gets very large. There are three types of behaviour that we shall wish to describe explicitly. These are sequeces that ted to ifiity ; sequeces that ted to mius ifiity ; sequeces that coverge to a real limit. 3 c mathcetre July 18, 2005
4 First we shall look at sequeces that ted to ifiity. We say a sequece teds to ifiity if, however large a umber, the sequece becomes greater tha that umber, ad stays greater. So if we plot a graph of a sequece tedig to ifiity, the the poits of the sequece will evetually stay above ay horizotal lie o the graph. ay umber Here are some examples of sequeces that ted to ifiity. At some poit this sequece will be greater tha ay umber we choose, ad stay greater, so the sequece teds to ifiity. ay umber the sequece ( 2 ) = 1 Eve though this sequece sometimes decreases, it will still evetually become greater, ad stay greater, tha ay umber. ay umber the sequece (1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6,... ) c mathcetre July 18,
5 Here are some sequeces that do ot ted to ifiity. For this sequece we obtai the secod term by addig 100 to the first term. The, to get the ext term, we add 50, ad the we add 25, ad so o. Each time we add half as much as we did before. Eve though this sequece starts by icreasig very quickly, it will ever be larger tha 200. So we ca choose a umber that will ever exceed. 200 the sequece (0, 100, 150, 175, ,... ) This sequece does get larger tha ay umber. But it does ot stay larger, because it always returs to zero. So we caot say that this sequece teds to ifiity. ay umber the sequece (0, 1, 0, 2, 0, 3, 0, 4, 0, 5,... ) There is some otatio we ca use if a sequece teds to ifiity. If the sequece (x ) teds to ifiity, we write either x as or lim (x ) =. Key Poit A sequece (x ) teds to ifiity if, however large a umber, the sequece will evetually become greater tha that umber, ad stay greater. 5 c mathcetre July 18, 2005
6 We shall ow look at sequeces that ted to mius ifiity. We say a sequece teds to mius ifiity if, however large a egative a umber, the sequece evetually becomes less tha that umber ad stays less tha it. So if we plot a graph of a sequece tedig to mius ifiity, the the poits of the sequece will evetually stay below ay horizotal lie o the graph. ay umber For example, this sequece teds to mius ifiity. ay umber the sequece ( 3 ) = 1 But this sequece does ot ted to mius ifiity. It does become less tha ay umber we choose. But it does ot stay less, because it always becomes positive agai. So we caot say that this sequece teds to mius ifiity. ay umber the sequece ( 1, 1, 2, 2, 3, 3,... ) c mathcetre July 18,
7 There is some otatio we ca use if a sequece teds to mius ifiity. If the sequece (x ) teds to mius ifiity, we write either x as or lim (x ) =. Key Poit A sequece (x ) teds to mius ifiity if, however large a egative umber, the sequece will evetually become less tha that umber, ad stay less. Fially we shall look at sequeces with real limits. We say a sequece teds to a real limit if there is a real umber, l, such that the sequece gets closer ad closer to it. We say l is the limit of the sequece. The sequece gets closer ad closer to l if, wheever we draw a iterval as arrow as aroud l, the sequece evetually gets trapped iside the iterval. l Notice that it does ot matter whether or ot the sequece evetually takes the value l. We just eed it to get as close as to l. This sequece gets as close as we like to zero. So we say the sequece teds to zero as ted to ifiity. l 1 the sequece ( ) = 1 7 c mathcetre July 18, 2005
8 This sequece teds to the limit 3. Eve though the values of the sequece sometimes move away from 3, they evetually stay withi ay iterval aroud 3 that we choose. The arrower the iterval, the further alog the sequece we might have to go before the values become trapped withi the iterval. 3 There is some otatio we ca use if a sequece teds to a real limit. If the sequece (x ) teds to the limt l, we write either x l as or lim (x ) = l. Key Poit A sequece (x ) teds to a real limit l if, however small a iterval aroud l that, the sequece will evetually take values withi that iterval, ad remai there. We say a sequece is diverget if it does ot coverge to a real limit. For example, if a sequece teds to ifiity or to mius ifiity the it is diverget. But ot all diverget sequeces ted to plus or mius ifiity. For example, the sequece (1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2,...) is diverget, but it does ot ted to either ifiity or to mius ifiity. A sequece like this, repeatig itself over ad over agai, is called a periodic sequece. Key Poit A sequece that does ot coverge to a real limit is called a diverget sequece. c mathcetre July 18,
9 Exercises Decide whether each of the followig sequeces teds to ifiity, teds to mius ifiity, teds to a real limit, or does ot ted to a limit at all. If a sequece teds to a real limit, work out what it is. ( ) 1. (2 ) =1 2. (1000 ) = (+ ) =1 5. Aswers ( si π 4 ) = ( 5 1 =1 ) 1. ifiity 2. mius ifiity ifiity 5. o limit 6. 5 =1 9 c mathcetre July 18, 2005
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