# The Limit of a Sequence

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 3 The Limit of a Sequece 3. Defiitio of limit. I Chapter we discussed the limit of sequeces that were mootoe; this restrictio allowed some short-cuts ad gave a quick itroductio to the cocept. But may importat sequeces are ot mootoe umerical methods, for istace, ofte lead to sequeces which approach the desired aswer alterately from above ad below. For such sequeces, the methods we used i Chapter wo t work. For istace, the sequece.,.9,.,.99,.,.999,... has as its limit, yet either the iteger part or ay of the decimal places of the umbers i the sequece evetually becomes costat. We eed a more geerally applicable defiitio of the limit. We abado therefore the decimal expasios, ad replace them by the approximatio viewpoit, i which the limit of {a } is L meas roughly a is a good approximatio to L, whe is large. The followig defiitio makes this precise. After the defiitio, most of the rest of the chapter will cosist of examples i which the limit of a sequece is calculated directly from this defiitio. There are limit theorems which help i determiig a limit; we will preset some i Chapter 5. Eve if you kow them, do t use them yet, sice the purpose here is to get familiar with the defiitio. Defiitio 3. The umber L is the limit of the sequece {a } if () give ǫ >, a ǫ L for. If such a L exists, we say {a } coverges, or is coverget; if ot, {a } diverges, or is diverget. The two otatios for the limit of a sequece are: lim {a } = L ; a L as. These are ofte abbreviated to: lim a = L or a L. Statemet () looks short, but it is actually fairly complicated, ad a few remarks about it may be helpful. We repeat the defiitio, the build it i three stages, listed i order of icreasig complexity; with each, we give its traslatio ito Eglish. 35

2 36 Itroductio to Aalysis Defiitio 3. lim a = L if: give ǫ >, a ǫ L for. Buildig this up i three succesive stages: (i) a L (a ǫ approximates L to withi ǫ); ( ) the approximatio holds for all a (ii) a L for ; ǫ far eough out i the sequece; (iii) give ǫ >, a L for ǫ (the approximatio ca be made as close as desired, provided we go far eough out i the sequece the smaller ǫ is, the farther out we must go, i geeral). The heart of the limit defiitio is the approximatio (i); the rest cosists of the if s, ad s, ad but s. First we give a example. Example 3.A Show lim + =, directly from defiitio 3.. Solutio. Accordig to defiitio 3., we must show: (2) give ǫ >, + ǫ for. We begi by examiig the size of the differece, ad simplifyig it: + = 2 + = 2 +. We wat to show this differece is small if. Use the iequality laws: 2 + < ǫ if + > 2 ǫ, i.e., if > N, where N = 2 ǫ ; this proves (2), i view of the defiitio (2.6) of for. The argumet ca be writte o oe lie (it s ugrammatical, but easier to write, prit, ad read this way): Solutio. Give ǫ >, + = 2 + < ǫ, if > 2 ǫ. Remarks o limit proofs.. The heart of a limit proof is i the approximatio statemet, i.e., i gettig a small upper estimate for a L. Ofte most of the work will cosist i showig how to rewrite this differece so that a good upper estimate ca be made. (The triagle iequality may or may ot be helpful here.) Note that i doig this, you must use ; you ca drop the absolute value sigs oly if it is clear that the quatity you are estimatig is o-egative. 2. I givig the proof, you must exhibit a value for the N which is lurkig i the phrase for. You eed ot give the smallest possible N; i example 3.A, it was 2/ǫ, but ay bigger umber would do, for example N = 2/ǫ. Note that N depeds o ǫ: i geeral, the smaller ǫ is, the bigger N is, i.e., the further out you must go for the approximatio to be valid withi ǫ.

3 Chapter 3. The Limit of a Sequece I Defiitio 3. of limit, the phrase give ǫ > has at least five equivalet forms; by covetio, all have the same meaig, ad ay of them ca be used. They are: for all ǫ >, for every ǫ >, for ay ǫ > ; give ǫ >, give ay ǫ >. The most stadard of these phrases is for all ǫ >, but we feel that if you are meetig () for the first time, the phrases i the secod lie more early capture the psychological meaig. Thik of a limit demo whose oly purpose i life is to make it hard for you to show that limits exist; it always picks upleasatly small values for ǫ. Your task is, give ay ǫ the limit demo hads you, to fid a correspodig N (depedig o ǫ) such that a ǫ L for > N. Remember: the limit demo supplies the ǫ; you caot choose it yourself. I writig up the proof, good mathematical grammar requires that you write give ǫ > (or oe of its equivalets) at the begiig; get i the habit ow of doig it. We will discuss this later i more detail; briefly, the reaso is that the N depeds o ǫ, which meas ǫ must be amed first. 4. It is ot hard to show (see Problem 3-3) that if a mootoe sequece {a } has the limit L i the sese of Chapter higher ad higher decimal place agreemet the L is also its limit i the sese of Defiitio 3.. (The coverse is also true, but more trouble to show because of the difficulties with decimal otatio.) Thus the limit results of Chapter, the Completeess Property i particular, are still valid whe our ew defiitio of limit is used. From ow o, limit will always refer to Defiitio 3.. Here is aother example of a limit proof, more tricky tha the first oe. Example 3.B Show lim ( + ) =. Solutio. We use the idetity A B = A2 B 2, which tells us that A + B (3) ( + ) = < ; give ǫ >, 2 < ǫ if 4 < ǫ2, i.e., if > 4ǫ 2. Note that here we eed ot use absolute values sice all the quatities are positive. It is ot at all clear how to estimate the size of + ; the triagle iequality is useless. Lie (3) is thus the key step i the argumet: the expressio must first be trasformed by usig the idetity. Eve after doig this, lie (3) gives a further simplifyig iequality to make fidig a N easier; just try gettig a N without this step! The simplificatio meas we do t get the smallest possible N; who cares?

4 38 Itroductio to Aalysis Questios 3.. Directly from the defiitio of limit (i.e., without usig theorems about limits you leared i calculus), prove that (a) + (b) cos a (a is a fixed umber) 2 + (c) 2 (d) 2 ( ) cf. Example 3.B: make 3 + a simplifyig iequality 2. Prove that, for ay sequece {a }, lim a = lim a =. (This is a simple but importat fact you ca use from ow o.) 3. Why does the defiitio of limit say ǫ >, rather tha ǫ? 3.2 The uiqueess of limits. The K-ǫ priciple. Ca a sequece have more tha oe limit? Commo sese says o: if there were two differet limits L ad L, the a could ot be arbitrarily close to both, sice L ad L themselves are at a fixed distace from each other. This is the idea behid the proof of our first theorem about limits. The theorem shows that if {a } is coverget, the otatio lima makes sese; there s o ambiguity about the value of the limit. The proof is a good exercise i usig the defiitio of limit i a theoretical argumet. Try provig it yourself first. Theorem 3.2A Uiqueess theorem for limits. A sequece a has at most oe limit: a L ad a L L = L. Proof. By hypothesis, give ǫ >, a ǫ L for, ad a ǫ L for. Therefore, give ǫ >, we ca choose some large umber k such that L ǫ a k ǫ L. By the trasitive law of approximatio (2.5 (8)), it follows that (4) give ǫ >, L 2ǫ L. To coclude that L = L, we reaso idirectly (cf. Appedix A.2). Suppose L L ; choose ǫ = 2 L L. We the have Remarks. L L < 2ǫ, by (4); i.e., L L < L L, a cotradictio.. The lie (4) says that the two umbers L ad L are arbitrarily close. The rest of the argumet says that this is osese if L L, sice they caot be closer tha L L.

5 Chapter 3. The Limit of a Sequece Before, we emphasized that the limit demo chooses the ǫ; you caot choose it yourself. Yet i the proof we chose ǫ = 2 L L. Are we blowig hot ad cold? The differece is this. Earlier, we were tryig to prove a limit existed, i.e., were tryig to prove a statemet of the form: give ǫ >, some statemet ivolvig ǫ is true. To do this, you must be able to prove the truth o matter what ǫ you are give. Here o the other had, we do t have to prove (4) we already deduced it from the hypothesis. It s a true statemet. That meas we re allowed to use it, ad sice it says somethig is true for every ǫ >, we ca choose a particular value of ǫ ad make use of its truth for that particular value. To reiforce these ideas ad give more practice, here is a secod theorem which makes use of the same priciple, also i a idirect proof. The theorem is obvious usig the defiitio of limit we started with i Chapter, but we are committed ow ad for the rest of the book to usig the ewer Defiitio 3. of limit, ad therefore the theorem requires proof. Theorem 3.2B {a } icreasig, L = lim a a L for all ; {a } decreasig, L = lim a a L for all. Proof. Both cases are hadled similarly; we do the first. Reasoig idirectly, suppose there were a term a N of the sequece such that a N > L. Choose ǫ = 2 (a N L). The sice {a } is icreasig, a L a N L > ǫ, for all N, cotradictig the Defiitio 3. of L = lim a. The K-ǫ priciple. I the proof of Theorem 3.2A, ote the appearace of 2ǫ i lie (4). It ofte happes i aalysis that argumets tur out to ivolve ot just ǫ but a costat multiple of it. This may occur for istace whe the limit ivolves a sum or several arithmetic processes. Here is a typical example. Example 3.2 Let a = + si +. Show a, from the defiitio. Solutio To show a is small i size, use the triagle iequality: + si + + si +. At this poit, the atural thig to do is to make the separate estimatios < ǫ, for > ǫ ; si + < ǫ, for > ǫ ; so that, give ǫ >, + si + < 2ǫ, for > ǫ. This is close, but we were supposed to show a < ǫ. Is 2ǫ just as good?

7 Chapter 3. The Limit of a Sequece 4 As for regular limits, to establish that lim{a } =, what you have to do is give a explicit value for the N cocealed i for, ad prove that it does the job, i.e., prove that a > M whe N. I geeral, this N will deped o M: the bigger the M, the further out i the sequece you will have to go for the iequality a > M to hold. As before, it is ot you who chooses the M; the limit demo does that, ad you have to prove the iequality i (6) for whatever positive M it gives you. Note also that eve though we are dealig with size, we do ot eed absolute values, sice a > M meas the a are all positive for. Oe should ot thik that ifiite limits are associated oly with icreasig sequeces. Cosider these examples, either of which is a icreasig sequece. Examples 3.3A Do the followig sequeces ted to? Give reasoig. (i) {a } =,, 2, 2, 3, 3, 4, 4,..., k, k,..., (k ); (ii) {a } =, 2,, 3,...,, k,..., (k ). { 5, eve; Solutio. (i) A formula for the -th term is a = ( + )/2, odd. This shows the sequece teds to sice (6) is satisfied: give M >, a > M if ( + )/2 > M; i.e., if > 2M. (ii) The secod sequece does ot ted to, sice (6) is ot satisfied for every give M: if we take M =, for example, it is ot true that after some poit i the sequece all a >, sice the term occurs at every odd positio i the sequece. Example 3.3B Show that {l }. Solutio. We use the fact that lx is a icreasig fuctio, that is, therefore, la > lb if a > b; give M >, l > l(e M ) = M if > e M. Questios 3.3. (a) Formulate a defiitio for lim a = : a teds to. (b) Prove l(/). 2. Which of these sequeces ted to? For those that do, prove it. (a) ( ) (b) si π/2 (c) (d) + cos 3. Prove: if a, the a is positive for large.

8 42 Itroductio to Aalysis 3.4 A importat limit. As a good opportuity to practice with iequalities ad the limit defiitio, we prove a importat limit that will be used costatly later o. Theorem 3.4 The limit of a. (7) lim a = Proof., if a > ;, if a = ;, if a <. We cosider the case a > first. Sice a >, we ca write a = + k, k >. Thus a = ( + k), which by the biomial theorem ( ) = + k + k 2 ( )( 2) + k k. 2! 3! Sice all the terms o the right are positive, (5) a > + k ; > M, for ay give M >, if > M/k, say. This proves that lim a = if a >, accordig to Defiitio 3.3. The secod case a = is obvious. For the third, i outlie the proof is: a < ( ) a > a. a Here the middle implicatio follows from the first case of the theorem. The last implicatio uses the defiitio of limit; amely, by hypothesis, ( ) give ǫ >, > for large; a ǫ by the reciprocal law of iequalities (2.) ad the multiplicatio law for, a < ǫ for large. Why did we begi by writig a = + k? Experimetally, you ca see that whe a >, but very close to (like a =.), a icreases very slowly at first whe raised to powers. This is the worst case, therefore, ad it suggests writig a i a form which shows how far it deviates from. The case a is ot icluded i the theorem; here the a alterate i sig without gettig smaller, ad the sequece has o limit. A formal proof of this directly from the defiitio of limit is awkward; istead we will prove it at the ed of Chapter 5, whe we have more techique. Questios 3.4. Fid (a) lim cos a; (b) lim l a, for a. 2. Suppose oe tries to prove the theorem for the case < a < directly, by writig a = k, where < k < ad imitatig the argumet give for the first

10 44 Itroductio to Aalysis 3.6 Some limits ivolvig itegrals. To broade the rage of applicatios ad get you thikig i some ew directios, we look at a differet type of limit which ivolves defiite itegrals. Example 3.6A Let a = (x 2 + 2) dx. Show that lim a =. The way ot to do this is to try to evaluate the itegral, which would just produce a uwieldy expressio i that would be hard to iterpret ad estimate. To show that the itegral teds to ifiity, all we have to do is get a lower estimate for it that teds to ifiity. Solutio. therefore, Thus We estimate the itegral by estimatig the itegrad. x for all x; (x 2 + 2) 2 for all x ad all. (x 2 + 2) dx 2 dx = 2. Sice lim 2 = by Theorem 3.4, the defiite itegral must ted to also: give M >, Example 3.6B Show lim (x 2 + 2) dx 2 M, for > log 2 M. (x 2 + ) dx =. Solutio. Oce agai, we eed a lower estimate for the itegral that is large. The previous argumet gives the estimate (x 2 + ) =, which is useless. However, it may be modified as follows. Sice x 2 + is a icreasig fuctio which has the value A = 5/4 at the poit x =.5 (ay other poit o (,) would do just as well), we ca say therefore, x 2 + A > for.5 x ; (x 2 + ) A for.5 x ; sice lim A = by Theorem 3.4, the defiite itegral must ted to also: give M >, (x 2 + ) dx.5 A dx = A 2 M, for large. Questios 3.6. By estimatig the itegrad, show that: 2. Show without itegratig that lim 3 x 2 + x dx. x ( x) dx =.

12 46 Itroductio to Aalysis Exercises 3.. Show that the followig sequeces have the idicated limits, directly from the defiitio of limit. si cos (a) lim (c) lim (e) = 3.2 lim 2 = (b) lim = (d) lim + 2 = 2 3 = 2. Prove that if a is a o-egative sequece, lim a = lim a =.. Prove that if a L ad b M, the a + b L + M. Do this directly from the defiitio 3. of limit. 2. Suppose {a } is a coverget icreasig sequece, ad lim a = L. Let {b } be aother sequece iterwove with the first, i.e., such that a < b < a + for all. Prove from the defiitio of limit that lim b = L also. 3. (a) Prove the sequece a = (b) Criticize the followig proof that its limit is : Give ǫ >, the for i =,2,3,..., we have < ǫ, if < ǫ, i.e., if > /ǫ. + i Addig up these iequalities for i =,..., gives therefore, < a < ǫ, for > /ǫ ; a ǫ, for. By the defiitio of limit ad the K-ǫ priciple, lim a =. ( 4. Prove that lim ) 2 =. + (Modify the icorrect argumet i the precedig exercise.) has a limit. 5. Let {a } be a coverget sequece of itegers, havig the limit L. Prove that it is evetually costat, that is, a = L for large. (Apply the limit defiitio, takig ǫ = /4, say. Why is it legal for you to choose the ǫ i this case?)

13 Chapter 3. The Limit of a Sequece For each sequece {a }, tell whether or ot a ; if so, prove it directly from Defiitio 3.3 of ifiite limit. (a) a = 2 (b) a = 2 cos π (c) a = (d) a =, 2, 3, 2, 3, 4, 3, 4, 5,... (e) (f) l l() 2. Prove: if a < b for, ad a, the b. Base the proof o Defiitio Prove: if {a }, the {a } is ot bouded above. (This is obvious ; the poit is to get practice i usig the defiitios to costruct argumets. Give a idirect proof; see Appedix A.2.) 3.4. Defie a recursively by a + = ca, where c <. Prove lim a =, usig Theorem 3.4 ad Defiitio Let a = r /!. Prove that for all r R, we have a. (If r, this is easy. If r >, it is more subtle. Compare two successive terms of the sequece, ad show that if, the a + is less tha half of a. The complete the argumet.) 3. Prove that if a >, the a /. (Hit: imitate the proof i theorem 3.4, but use a differet term i the biomial expasio.) 4. Prove that a if < a <, usig the result i the precedig exercise. 5. Prove that lima / =, if a >. (Here a / meas the real positive -th root of a. Nothig is said about a <, sice the a / is ot a real umber if is eve. Note how {a / } ad {a } have opposite behavior as sequeces. As we take successive -th roots, all positive umbers approach ; i cotrast, as we take successive -th powers, all positive umbers recede from.) (a) Cosider first the case a >. For each, put a / = + h, ad show that h, by reasoig like that i Theorem 3.4. (b) Now cosider the case a <. If a <, the /a > ; use this ad the defiitio of limit to deduce this case from the previous oe. (Use oly the defiitio of limit i this chapter, ot other obvious facts about limits.) 3.6. Modelig your argumets o the two examples give i this sectio, prove the followig without attemptig to evaluate the itegrals explicitly. (a) lim 2 l x dx = (b) lim 3 2 l x dx =

14 48 Itroductio to Aalysis 3.7. Show that lim the itegral explicitly. ( x 2 ) dx =, without attemptig to evaluate Problems 3- Let {a } be a sequece ad {b } be its sequece of averages: b = (a a )/ (cf. Problem 2-). (a) Prove that if a, the b. (Hit: this uses the same ideas as example 3.7. Give ǫ >, show how to break up the expressio for b ito two pieces, both of which are small, but for differet reasos.) (b) Deduce from part (a) i a few lies without repeatig the reasoig that if a L, the also b L. 3-2 To prove a was large if a >, we used Beroulli s iequality : ( + h) + h, if h. We deduced it from the biomial theorem. This iequality is actually valid for other values of h however. A sketch of the proof starts: ( + h) 2 = + 2h + h 2 + 2h, sice h 2 for all h; ( + h) 3 = ( + h) 2 ( + h) ( + 2h)( + h), by the previous case, = + 3h + 2h 2, + 3h. (a) Show i the same way that the truth of the iequality for the case implies its truth for the case +. (This proves the iequality for all by mathematical iductio, sice it is trivially true for =.) (b) For what h is the iequality valid? (Try it whe h = 3, = 5.) Recocile this with part (a). 3-3 Prove that if a is a bouded icreasig sequece ad lima = L i the sese of Defiitio.3A, the lima = L i the sese of Defiitio Prove that a coverget sequece {a } is bouded. 3-5 Give ay c R, prove there is a strictly icreasig sequece {a } ad a strictly decreasig sequece {b }, both of which coverge to c, ad such that all the a ad b are (i) ratioal umbers; (Theorem 2.5 is helpful.) (ii) irratioal umbers.

15 Chapter 3. The Limit of a Sequece 49 Aswers 3.. We write these up i four slightly differet styles; take your pick. (a) Give ǫ >, + = +, (b) Give ǫ >, (c) Give ǫ >, which is < ǫ if > /ǫ, or > /ǫ. cos a, sice cos x for all x; ad / < ǫ if > /ǫ = 2 2, 2 < ǫ if 2 > 2/ǫ, or > 2/ǫ +. (d) Give ǫ >, 3 + < < ǫ if > /ǫ. 2. lim a = meas: give ǫ >, a < ǫ for. lim a = meas: give ǫ >, a < ǫ for. But these two statemets are the same, sice a = a = a = a. 3. If the limit demo were allowed to give you ǫ =, the sice is the same as equality =, it would have to be true that a = L for ; i other words, ay sequece which had the limit L would from some poit o have to be costat ad equal to L. This would be too restricted a otio of limit. (The sequeces which do behave this way are said to be evetually costat ; cf. Exercise 3.2/5.) 3.2. L L ǫ = L L /2 >, which is essetial (cf. Questio 3./3). 2. (a) Give ǫ >, a L < ǫ for > N, say. Therefore, so we re doe by the K-ǫ priciple. ca cl = c a L < c ǫ for > N, (b) Give ǫ >, a L < ǫ/ c for > N, say. Therefore, ca cl = c a L < c ǫ/ c = ǫ for > N. 3. Give ǫ >, (triagle iequality) < ǫ + ǫ, if + > /ǫ, ad > 2/ǫ; so we re doe by the K-ǫ priciple. < 2ǫ, if > + 2/ǫ;

16 5 Itroductio to Aalysis (a) Give M <, a < M for. (The sigs ca be omitted.) (b) l(/) = l < M if l > M, i.e., if > e M. 2. (a) o; alterate terms are egative; (b) o; alterate terms are ; (c) yes; give M >, > M if > M 2 ; (d) yes; give M >, + cos > > M, if > M Give M, a > M for. Take M = : a > for.. (a) limit is, except: limit is if a = 2π, o limit if a = (2 + )π. {, if a < e; (b) limit is:, if a = e;, if a > e. 2. Sice the terms alterate i sig, oe caot get a iequality after droppig most of the terms to simplify the expressio. Basically, a is small ot because the idividual terms are small, but because they cacel each other out. Thus the triagle iequality caot help either. So this approach does t lead to a usable estimatio that would show a is small Over the iterval x, x 2 + 2; 2 x By stadard reasoig (see Example 2.2B for istace), 3 x2 + x ; therefore its itegral over the iterval [,] of legth also lies betwee these bouds. 2. O [,], the maximum of x( x) occurs at x = /2, therefore give ǫ >, x( x) /4 o [,]; therefore, deotig the itegral by I, we see that which proves that limi =. x ( x) dx (/4) < ǫ, if 4 > /ǫ ; give ǫ >, I < ǫ if > l(/ǫ)/l 4,

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### Sequences II. Chapter 3. 3.1 Convergent Sequences

Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

### Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

### Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

### Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Sequences and Series

CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

### 4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

### Properties of MLE: consistency, asymptotic normality. Fisher information.

Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

### Section 11.3: The Integral Test

Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

### 4.3. The Integral and Comparison Tests

4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

### Convexity, Inequalities, and Norms

Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

### The second difference is the sequence of differences of the first difference sequence, 2

Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

### TAYLOR SERIES, POWER SERIES

TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the

### 8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

### Asymptotic Growth of Functions

CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

### Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University

Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced

INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

### Approximating the Sum of a Convergent Series

Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece

### Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

### Soving Recurrence Relations

Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

### BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

### 1. C. The formula for the confidence interval for a population mean is: x t, which was

s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

### 1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

### Taylor Series and Polynomials

Taylor Series ad Polyomials Motivatios The purpose of Taylor series is to approimate a fuctio with a polyomial; ot oly we wat to be able to approimate, but we also wat to kow how good the approimatio is.

### The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

### Lesson 17 Pearson s Correlation Coefficient

Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

### Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

### CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

### Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

### Class Meeting # 16: The Fourier Transform on R n

MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

### Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

### AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

### Chapter 5: Inner Product Spaces

Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

### FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

### Sum and Product Rules. Combinatorics. Some Subtler Examples

Combiatorics Sum ad Product Rules Problem: How to cout without coutig. How do you figure out how may thigs there are with a certai property without actually eumeratig all of them. Sometimes this requires

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

### WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

### Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

### a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x

0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio

### 3. Covariance and Correlation

Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

### Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

### Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### Building Blocks Problem Related to Harmonic Series

TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

### Math 113 HW #11 Solutions

Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

### Department of Computer Science, University of Otago

Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

### CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

### Chapter 7 Methods of Finding Estimators

Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

### THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

### AP Calculus AB 2006 Scoring Guidelines Form B

AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success

### {{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

### Math C067 Sampling Distributions

Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

### CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

### Recursion and Recurrences

Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

### Mathematical goals. Starting points. Materials required. Time needed

Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

### Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

### The Field Q of Rational Numbers

Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

### AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

### 1.3 Binomial Coefficients

18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

### Section 1.6: Proof by Mathematical Induction

Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems

### Chapter One BASIC MATHEMATICAL TOOLS

Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

### Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

### Unit 20 Hypotheses Testing

Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect

### x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### 1 Introduction to reducing variance in Monte Carlo simulations

Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

### Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

### CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

### 7. Sample Covariance and Correlation

1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### Problem Set 1 Oligopoly, market shares and concentration indexes

Advaced Idustrial Ecoomics Sprig 2016 Joha Steek 29 April 2016 Problem Set 1 Oligopoly, market shares ad cocetratio idexes 1 1 Price Competitio... 3 1.1 Courot Oligopoly with Homogeous Goods ad Differet

### Section 8.3 : De Moivre s Theorem and Applications

The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =

### Lesson 15 ANOVA (analysis of variance)

Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

### Estimating the Mean and Variance of a Normal Distribution

Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

### Standard Errors and Confidence Intervals

Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

### Solving Inequalities

Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK-12

### Chapter 14 Nonparametric Statistics

Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

### Using Excel to Construct Confidence Intervals

OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio

### 7.1 Finding Rational Solutions of Polynomial Equations

4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

### Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

### Overview of some probability distributions.

Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### NOTES ON INEQUALITIES FELIX LAZEBNIK

NOTES ON INEQUALITIES FELIX LAZEBNIK Order ad iequalities are fudametal otios of moder mathematics. Calculus ad Aalysis deped heavily o them, ad properties of iequalities provide the mai tool for developig

### 5.3. Generalized Permutations and Combinations

53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible

### SEQUENCES AND SERIES CHAPTER

CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for \$,200 o a istallmet pla, they agreed to pay \$00 each moth util the cost of the computer plus iterest had bee paid The iterest each