MA131 - Analysis 1. Workbook 7 Series I

Transcription

1 MA3 - Aalysis Workbook 7 Series I Autum 008 Cotets 4 Series 4. Defiitios Geometric Series The Harmoic Series Basic Properties of Coverget Series Boudedess Coditio Null Sequece Test Compariso Test * Applicatio - What is e? *

2 4 Series 4. Defiitios We saw i the last booklet that decimal expasios could be defied i terms of sequeces of sums. Thus a decimal expasio is like a ifiite sum. This is what we shall be lookig at for the rest of the course. Our first aim is to fid a good defiitio for summig ifiitely may umbers. The we will ivestigate whether the rules for fiite sums apply to ifiite sums. Exercise x =. What has goe wrog with the followig argumet? Try puttig If S = + x + x +..., the xs = x + x + x , so S xs =, ad therefore S = x. If the argumet were correct the we could put x = to obtai the sum of the series as /. But the same series could also be be thought of as with a sum of 0, or as ) + ) + ) ) + + ) + + ) +... with a sum of. This shows us that great care must be exercised whe dealig with ifiite sums. We shall repeatedly use the followig coveiet otatio for fiite sums: give itegers 0 m ad umbers a : = 0,,... ) we defie a k = a m + a m+ + + a k=m 0 Example = k= Exercise Express the followig sums usig the otatio: k Exercise 3 Show that k= a k = a k.

3 Exercise 4. By decomposig /rr + ) ito partial fractios, or by iductio, prove that ) = +.. If Write this result usig otatio. s = r= rr + ), prove that s ) as. This result could also be writte as r= rr + ) =. A series is a expressio of the form = a = a + a + a 3 + a As yet, we have ot defied what we mea by such a ifiite sum. To get the ball rollig, we cosider the partial sums of the series: s = a s = a + a s 3 = a + a + a 3.. s = a + a + a a To have ay hope of computig the ifiite sum a + a + a , the the partial sums s should represet closer ad closer approximatios as is chose larger ad larger. This is just a iformal way of sayig that the ifiite sum a + a + a ought to be the limit of the sequece of partial sums. Defiitio Cosider the series = a = a + a + a with partial sums s ), where We say: s = a + a + a a =. = a coverges if s ) coverges. If s S the we call S the sum of the series ad we write = a = S.. = a diverges if s ) does ot coverge. 4. = a diverges to mius ifiity if s ) teds to mius ifiity. i= 3. = a diverges to ifiity if s ) teds to ifiity. a i Serious Sums The problem of how to deal with ifiite sums vexed the aalysis of the early 9th cetury. Some said there was t a problem, some preteded there was t util icosistecies i their ow work bega to uerve them, ad some said there was a terrible problem ad why would t ayoe liste? Evetually, everyoe did. Double Trouble There are two sequeces associated with every series P = a: the sequece a) ad the sequece of partial sums s ) = `P i= ai. Do ot get these sequeces cofused! Series Need Sequeces Notice that series covergece is defied etirely i terms of sequece covergece. We have t spet six weeks workig o sequeces for othig!

4 We sometimes write the series = a simply as a. Example Cosider the series = = The sequece of partial sums is give by s = = ) ) ) = Clearly s. It follows from the defiitio that = = = = We could express the argumet more succictly by writig ) ) = lim k = lim = k= Example Cosider the series = ) = Here we have the partial sums: s = a = s = a + a = 0 s 3 = a + a + a 3 = s 4 = a + a + a 3 + a 4 = ad we ca see at oce that the sequece s ) =, 0,, 0,... does ot coverge. Assigmet Look agai at the series =. Plot o two small separate graphs both the sequeces a ) = ) ad s ) = ) k=. k Assigmet Fid the sum of the series ) = 0. Dummy Variables Make careful ote of the way the variables k ad appear i this example. They are dummies - they ca be replaced by ay letter you like. Frog Hoppig Heard about that frog who hops halfway across his pod, ad the half the rest of the way, ad the half that, ad half that, ad half that...? Is he ever goig to make it to the other side? Assigmet 3 Reread your aswer to exercise 4 ad the write out a full proof that = + ) = = Assigmet 4 Show that the series diverges to +. [Hit: Calculate the partial sums s, s 3, s 6, s 0,....] 3

5 4. Geometric Series Theorem Geometric Series The series =0 x is coverget if x < ad the sum is x. It is diverget if x. Exercise 5 ) =0 = coverges. =0.), =0 ) ad =0 3) = all diverge. Assigmet 5 Prove the theorem [Hit: Use the GP formula to get a formula for s ]. 4.3 The Harmoic Series The series = = is called the Harmoic Series. The followig groupig of its terms is rather cuig: + }{{} / }{{ 4 } }{{ 8 } }{{ 6 } / / / Assigmet 6 Prove that the Harmoic Series diverges. Structure your proof as follows:. Let s = k= k be the partial sum. Show that s s + for all. Use the idea i the cuig groupig above).. Show by iductio that s + 3. Coclude that = diverges. for all. Assigmet 7 Give, with reasos, a value of N for which N Basic Properties of Coverget Series Some properties of fiite sums are easy to prove for ifiite sums: Theorem Sum Rule for Series Suppose = a ad = b are coverget series. The, for all real umbers c ad d, = ca + db ) is a coverget series ad ca + db ) = c a + d = = = b GP Cosultatio How could you ever forget that a + ax + ax + + ax = a whe x? x x Harmoic History There are other proofs that the Harmoic Series is diverget, but this is the origial. It was cotributed by the Eglish mediaeval mathematicia Nicholas Oresme 33-38) who also gave us the laws of expoets: x m.x = x m+ ad x m ) = x m. Coflictig Covergece You ca see from this example that the covergece of a ) does ot imply the covergece of P = a. 4

6 Proof. i= ca + db ) = c i= a i) + d i= b i) c = a + d = b Theorem Shift Rule for Series Let N be a atural umber. The the series = a coverges if ad oly if the series = a N+ coverges. Example diverget. We showed that = is diverget. It follows that = + is Assigmet 8 Prove the shift rule. 4.5 Boudedess Coditio If the terms of a series are all o-egative, the we shall show that the boudedess of its partial sums is eough to esure covergece. Theorem Boudedess Coditio Suppose a 0. The = a coverges if ad oly if the sequece of partial ) sums s ) = j= a j is bouded. Assigmet 9 Prove this result. Your proof must use the axiom of completeess or oe of its cosequeces - make sure you idicate where this occurs. 4.6 Null Sequece Test Assigmet 0. Prove that if = a coverges the the sequece a ) teds to zero. Hit: Notice that a + = s + s ad use the Shift Rule for sequeces.). Is the coverse true: If a ) 0 the = a coverges? We have proved that if the series = a coverges the it must be the case that a ) teds to zero. The cotrapositive of this statemet gives us a test for divergece: Theorem Null Sequece Test If a ) does ot ted to zero, the = a diverges. 5 Red Alert The Null Sequece Test is a test for divergece oly. You ca t use it to prove series covergece.

7 Example The sequece ) does ot coverge to zero, therefore the series = diverges. 4.7 Compariso Test The ext test allows you to test the covergece of a series by comparig its terms with those of a series whose behaviour you already kow. Theorem Compariso Test Suppose 0 a b for all atural umbers. If b coverges the a coverges ad = a = b. Example You showed i assigmet 3 that +) coverges. Now 0 +) +). It follows from the Compariso Test that +) also coverges ad via the Shift Rule that the series = coverges. Exercise 6 Give a example to show that the test fails if we allow the terms of the series to be egative, i.e. if we oly demad that a b. Like the Boudedess Coditio, you ca oly apply the Compariso Test ad the other tests i this sectio) if the terms of the series are oegative. Assigmet Prove the Compariso Test [Hit: Cosider the partial sums of both b ad a ad show that the latter is icreasig ad bouded]. Exercise 7 Check that the cotrapositive of the statemet: If b coverges the a coverges. gives you the followig additioal compariso test: Corollary Compariso Test Suppose 0 a b. If a diverges the b diverges. Examples. Note 0. We kow diverges, so diverges too.. To show that diverges, otice that that diverges, therefore + + diverges. + + =. We kow 6

8 Assigmet Use the Compariso Test to determie whether each of the followig series coverges or diverges. I each case you will have to thik of a suitable series with which to compare it. i) ii) si x 4.8 * Applicatio - What is e? * iii) Over the years you have o doubt formed a workig relatioship with the umber e, ad you ca say with cofidece ad the aid of your calculator) that e.78. But that is ot the ed of the story. Just what is this e umber? To aswer this questio, we start by ivestigatig the mysterious series =0!. Note that we adopt the covetio that 0! =. Way To Go Stare deeply at each series ad try to fid a simpler series whose terms are very close for large. This gives you a good idea which series you might hope to compare it with, ad whether it is likely to be coverget or diverget. For istace the terms of the series P + are like those of the + series P for large values of, so we would expect it to diverge. Assigmet 3 Cosider the series =0! = + +! + 3! + 4! +... ad its partial sums s = k! = + +! + 3! + 4! !.. Show that the sequece s ) is icreasig.. Prove by iductio that! for > Use the compariso test to coclude that =0! coverges. Now here comes the Big Defiitio we ve all bee waitig for...!!! Defiitio e := = )! = + +! + 3! + 4! +... Recall that i the last workbook we showed that lim + ) exists. We ca ow show, with some rather delicate work, that this limit equals e. Assigmet 4 Use the Biomial Theorem to show that + ) = ) k! )... ) k ) k! Biomial Theorem For all real values x ad y ad iteger =,,...! X x + y) = x k y k k Coclude that lim + ) e. where ` k = here we use 0! =.!. k! k)! Note 7

9 We ow aim to show lim + ) e. The first step is to show that for all m ad + ) m+ By the Biomial theorem, + ) m+ = +m m k! + m)! k! + m k)! k m + m)! k! + m k)! where we have throw away the last terms of the sum. So + ) m+ which proves equatio ). m m k + m) + m )... + m k + ) k! k k! Assigmet 5 Cosider the iequality i equatio ). Fix m ad let to coclude that lim + m ) k!. The let m ad show that lim + ) e. ) You have proved: Theorem e = lim + ) Assigmet 6. Show that ). + = +/) ad hece fid lim +). Use the shift rule to fid lim ). The last exercise i this booklet is the proof that e is a irratioal umber. The proof uses the fact that the series for e coverges very rapidly ad this same idea ca be used to show that may other series also coverge to irratioal umbers. Theorem e is irratioal. 8

10 Assigmet 7 Prove this result by cotradictio. Structure your proof as follows: This exercise will ot be marked for credit.. Suppose e = p q ad show that e q+ i= some positive iteger k. i )! = p q q+ i= i )! = k q! for. Show that e q+ i= i )! = q+)! + q+)! + q+3)! + < ) ad derive a cotradictio to part. q! Check Your Progress By the ed of this Workbook you should be able to: Uderstad that a series coverges if ad oly if its partial sums coverge, i which case = a = lim i= a i). Write dow a list of examples of coverget ad diverget series ad justify your choice. Prove that the Harmoic Series is diverget. State, prove, ad use the Sum ad Shift Rules for series. State, prove, ad use the Boudedess Coditio. Use ad justify the Null Sequece Test. Describe the behaviour of the Geometric Series = x. State, prove ad use the Compariso Theorem for series. Justify the limit e = lim + ) startig from the defiitio e = =0!. Prove that e is irratioal. 9