Sequences, Series, and All That
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1 Chapter Te Sequeces, Series, ad All That. Itroductio Suppose we wat to compute a approximatio of the umber e by usig the Taylor polyomial p for f ( x) = e x at a =. This polyomial is easily see to be 3 x x x p ( x) = + x K +. 6! We could ow use p ( ) as a approximatio to e. We ow from the previous chapter that the error is give by e + e p ( ) = ( + )!, where < ξ <. Assume we ow that e <3, ad we have the estimate ξ 3 e p ( ) ( + )!. Meditate o this error estimate. It tells us that we ca mae this error as small as we lie by choosig sufficietly large. This is expressed formally by sayig that the limit of p ( ) as becomes ifiite is e. This is the idea we shall study i this chapter.. Sequeces A sequece of real umbers is simply a fuctio from a subset of the oegative itegers ito the reals. If the domai is ifiite, we say the sequece is a ifiite sequece. (Guess what a fiite sequece is.) We shall be cocered oly with ifiite sequeces, ad so the modifier will usually be omitted. We shall also almost always cosider sequeces i which the domai is either the etire set of oegative or positive itegers. There are several otatioal covetios ivolved i writig ad talig about sequeces. If f : Z + R, it is customary to deote f ( ) by f by ( f ). (Here Z + deotes the positive itegers.) Thus, for example,, ad the sequece itself is the sequece.
2 f defied by f ( ) =. The fuctio values f are called terms of the sequece. Frequetly oe sees a sequece described by writig somethig lie 4,, 9, K,, K. This is simply aother way of describig the sequece ( ). Let ( a ) be a sequece ad suppose there is a umber L such that for ay ε >, there is a iteger such that a L < ε for all >. The L is said to be a limit of the sequece, ad ( a ) is said to coverge to L. This is usually writte lim a = L. ow, what does this really mea? It says simply that as gets big, the terms of the sequece get close to L. I hope it is clear that is a limit of the sequece. From the discussio i the Itroductio to this chapter, it should be reasoably clear that a limit of the sequece K+ is e. 6! The graph of a sequece is pretty dreary compared with the graph of a fuctio whose domai is a iterval of reals, but evertheless, a loo at some pictures ca help uderstad some of these defiitios. Suppose the sequece ( a ) coverges to L. Loo at the graph of ( a ) : The fact that L is a limit of the sequece meas that for ay ε >, there is a so that to the right of, all the spots are i the strip of width ε cetered at L. Exercises.
3 . Prove that a sequece ca have at most oe limit (We may thus spea of the limit of a sequece.).. Give a example of a sequece that does ot have a limit. Explai. 3. Suppose the sequece ( a ) = a, a, a, K coverges to L. Explai how you ow that the sequece ( a ) a, a, a, K also coverges to L. + 5 = Fid the limit of the sequece 3, or explai why it does ot coverge. 5. Fid the limit of the sequece 3 + 7, or explai why it does ot coverge. 6. Fid the limit of the sequece coverge , or explai why it does ot Fid the limit of the sequece log, or explai why it does ot coverge..3 Series Suppose ( a ) is a sequece. The sequece ( a + a + K+ a ) is called a series. It is a little eater to write if we use the usual summatio otatio: example of such a thig previously; viz., = a. We have see a = 6 K.!! = It is usual to replace lim a = by = a. Thus, oe would, for example, write e = =!..3
4 Oe also frequetly sees the limit = a writte as a + a + K+ a + K. Ad oe more word of warig. Some poor misguided souls also use = a to stad simply for the series = a. It is usually clear whether the series or the limit of the series is meat, but it is evertheless a offesive practice that should be ruthlessly ad brutally suppressed. Example Let's cosider the series = = K. Let Thus S = K +. The 4 8 = K S S = S S =. + S This maes it quite easy to see that lim =, or lims =. I other words, =. = Observe that for series = a to coverge, it must be true that lim a =. To see this, suppose L = = a, ad observe that a = a a = =. Thus, lim a = lima a = lim a lim a = = = = = L L =..4
5 I other words, if lim a, the the series = a does ot have a limit. Aother Example Cosider the series =. First, ote that lim =. Thus we do ot ow that the series does ot coverge; that is, we still do't ow aythig. Loo at the followig picture: The curve is the graph of y =. Observe that the area uder the "stairs" is simply x. = ow covice yourself that to x = +. I other words, = is larger tha the area uder the curve y = from x = x = + > x dx = log( + )..5
6 We ow that log( + ) ca be made as large as we wish by choosig sufficietly large. Thus = ca be made as large as we wish by choosig sufficietly large. From this it follows that the series = does ot have a limit. (This series has a ame. It is called the harmoic series. ) The method we used to show that the harmoic series does ot coverge ca be used o may other series. We simply cosider a picture lie the oe above. Suppose we have a series = a such that a > for all. Suppose f is a decreasig fuctio such that f ( ) = for all. The if the limit lim f ( x) dx a R R does ot exist, the series is diverget. Exercises 8. Fid the limit of the series =, or explai why it does ot coverge Fid the limit of the series = 5, or explai why it does ot coverge Fid a value of that will isure that K+ >. 3 + θ. Let θ. Prove that si θ = ( ) ( + )! =. [Hit: p + ( θ) = ( ) = θ + ( + )! the fuctio f ( θ) = siθ at a =.] is the Taylor polyomial of degree < + for.6
7 . Suppose we have a series a such that a > for all, ad suppose f is a = decreasig fuctio such that f ( ) = for all. Show that if the limit a lim f ( x) dx R R exists, the the series is coverget. 3. a) Fid all p for which the series p = coverges. b) Fid all p for which the series i a) diverges..4 More Series Cosider a series = a i which a for all. This is called a positive series. Let = b be aother positive series. Suppose that b a for all >, where is simply some iteger. ow suppose further that we ow that = a coverges. This tells us all about the series = b. Specifically, it tells us that this series also coverges. Let's see why that is. First ote the obvious: = b coverges if ad oly if = b coverges. ext, observe that for all, we have b a follows at oce that lim b exists. = = =, from which it Example.7
8 What about the covergece of the series = ? Observe first that <. The observe that the series coverges because 3 = lim R lim x dx = + =. Thus 3 R 3R 3 3 R = coverges. + 4 Suppose that, as before, = a ad = b are positive series, ad b a for all >, where is some umber. This time, suppose we ow that = b is diverget. The it should ot be too hard for you to covice yourself that diverget, also. = a must be Exercises Which of the followig series are coverget ad which are diverget? Explai your aswers. 4. = e = 6. = log 7. = +.8
9 .5 Eve More Series We loo at oe more very ice way to help us determie if a positive series has a limit. Cosider a series = a, ad suppose a > for all. ext suppose the sequece a a + is coverget, ad let r a lim a + =. The umber r tells us almost everythig about the covergece of the series see about it. = a. Let's First, suppose that r <. The the umber ρ = r + r is positive ad less tha. For all sufficietly large, we ow that a a a a ρ for all. Thus + Loo ow at the series a a ρ a ρ a ρ K a ρ. + = a ρ = + ρ + ρ + ρ ρ. I other words, there is a so that ( a ( K )). This oe coverges because the Geometric series = ρ coverges (Recall that < ρ <.). It ow follows from the previous sectio that our origial series a limit. = has a A similar argumet should covice you that if r >, the the series ot have a limit. = a does The "method" of the previous sectio is usually called the Compariso Test, while that of this sectio is usually called the Ratio Test..9
10 Exercises Which of the followig series are coverget ad which are diverget? Explai your aswers. 8. =! 9. + = = 3. = ( + + ). = 4 3 ( + + ) 5.6 A Fial Remar The "tests" for covergece of series that we have see so far all depeded o the series havig positive terms. We eed to say a word about the situatios i which this is ot ecessarily the case. First, if the terms of a series true that a a for all, the lim + a = a alterate i sig, ad if it is = is sufficiet to isure covergece of the series. This is ot too hard to see meditate o it for a while. The secod result is a bit harder to see, ad we'll just put out the result as the word, asig that you accept it o faith. It says simply that if the series a = coverges, the so also does the series = a. Thus, faced with a arbitrary series a, we =.
11 may uleash out arseal of tests o the series a. If we fid this oe to be coverget, the the origial series is also coverget. If, of course, this series turs out ot to be coverget, the we still do ot ow about the origial series. =.
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