MATH SECTION NOTES EVAN WARNER. March 9.. Admiistrative miscellay. These weekly sectios will be for some review ad may example problems, i geeral. Attedace will be take as per class policy. We will be usig playig cards for immediate feedback; black meas yes or true, red meas o or false, ad the back of the card meas I have o idea (but this is a cop-out!). I geeral for admiistrative thigs the course website, http://web.staford. edu/class/math/, is quite comprehesive. These otes will be posted at http: //math.staford.edu/~ebwarer/... Sequeces. For our purposes, a sequece is just a list of umbers, which we otate with somethig like {a }, or if we wat to be pedatic about idices, {a } =. (The startig idex, usually zero or oe, is just a matter of covetio ad is usually chose so as to make the otatio i ay particular situatio as simple as possible.) Ofte sequeces are give by explicit formulas. For example, a =, defies the sequece,, 3, 4, 5,..., while a =, defies the sequece, 4, 9, 6, 5,.... Eve better tha writig out the first few terms, we ca graph sequeces just as we graph fuctios (the fuctios that we are familiar with are just maps from the real umbers to the real umbers; sequeces are maps from the atural umbers to the real umbers so here the x-axis is just a buch of poits goig off to the right). There s o reaso a sequece has to be give by a explicit formula (whatever that meas). Aother way of uiquely specifyig a sequece is by a recurrece relatio; the usual example is the Fiboacci umbers, which are defied as follows: F 0 =, F =, ad F = F + F for all. That is, we fix the first two values ad declare that each subsequet value is the sum of the previous two. This clearly specifies the sequece,,, 3, 5, 8, 3,.... (I fact, i this case oe ca without too much difficulty derive a explicit formula for the Fiboacci sequece, but with more complicated recurrece relatios this eed ot be the case. Ad for some purposes a recurrece relatio may be more useful tha a explicit formula.) A sequece {a } has a limit C if as gets larger ad larger, a gets closer ad closer to C. (A more precise defiitio is i sectio 8. of the text, but right ow we re just buildig ituitio.) We write lim a to deote this limit, if it exists. Of course it may ot, which makes the above otatio occasioally slightly frustratig. Here are some examples:
EVAN WARNER lim = 0 because as gets larger ad larger the square of its reciprocal gets closer ad closer to zero. lim does ot exist because there is o real umber such that as gets larger ad larger it gets closer ad closer to that umber. However, this limit does fail to exist i a particular way: as gets larger ad larger, the sequece evetually grows larger tha ay value (ad stays there), so i this situatio we write lim =. lim ( ) does ot exist: it oscillates forever betwee the values ad, ad so ever gets close to ay oe value. lim si(π) does exist; if we evaluate the sequece o ay we get zero ad the limit of the zero sequece is zero! So while the real-valued fuctio si(πx) oscillates ad cosequetly does ot have a limit as x, the sequece si(π) does have a limit as because it just so happes that all of the oscillatio happes away from the itegers. (Note the cofusig otatio, which does t properly distiguish betwee sequeces ad fuctios! Usually the use of a letter like i, j, k, l, m, is a sigal that we have a sequece, while a letter like t, u, v, x, y, z is a sigal that we are cosiderig a real-valued fuctio.).3. Series. We use the above cocept of a limit to defie what we mea by the sum of ifiitely may real umbers, which is ot a priori at all well-defied. Namely, we defie the sum of the sequece {a } =, which is usually otated a, = to be the limit of the sequece of partial sums. The sequece of partial sums is the sequece a, a + a, a + a + a 3, a + a + a 3 + a 4,...; it is the sequece whose th etry is the sum of the first terms (which is well-defied; we certaily kow how to add fiitely may umbers together!). A ifiite sum like this is called a series. Because limits of sequeces might ot exist, the value of a series might ot exist. Here are some examples: = 0 is, by defiitio, the limit of the sequece of partial sums, which is 0, 0 + 0, 0 + 0 + 0, 0 + 0 + 0 + 0,.... Obviously all terms of this sequece are zero, so the limit is also zero. Thus uder our defiitio of the sum of a sequece, ifiitely may zeroes sum to zero. = is the limit of the sequece, +, + +, + + +,..., which is just the sequece,, 3, 4,..., which has o limit (we say goes to ifiity). Therefore this sum is ot defied (ad we could say it is ifiite). = ( ) has partial sums, + = 0, + =, ad so o. That is, the sequece of partial sums oscillates forever betwee ad 0, ad cosequetly has o limit, so this sum is ot defied. I the 9:30 sectio, we had time to sketch a argumet that = does exist, eve if we ca t easily figure out what the precise value is. We will come back to this i more detail later.
MATH SECTION NOTES 3. April 5.. Series ad decimals. We should ow be moderately familiar with geometric series, which are oe of the oly kids of series we ca evaluate exactly. As a example, let s calculate S = = 3 0. The usual formula for the sum of a geometric series (proved i lecture, ad worth reviewig!) is r =, r <. r =0 If r it is easy to see that the series does ot coverge. Here we wat to apply it to a series startig at =, so we peel off the first term, which is r 0 = : r = + r, so = =0 = r = r = r r. Back to our example: we ca pull out the costat multiple of 3 (for the same reaso we ca pull out a costat multiple from a itegral: both are defied i terms of limits, ad we ca pull out costats from limits i the same way). So, usig the formula we derived above, S = 3 = 0 = 3 = ( ) = 3 0 0 0 = 3 9 = 3. Great! This sort of calculatio should get to the poit where it is quick ad easy. But here s a eve quicker way to see what the sum is. Let s write the partial sums i decimal otatio. We have S = 0.3, S = 0.33, S 3 = 0.333, ad so o. Clearly, i the limit we should have S = 0.3 = 3, where we recogize the decimal expasio i questio as precisely oe third. The uderlyig lesso here is that decimals really are a way of writig a real umber as a very particular kid of series! A lot of the ituitio we have for decimals is really ituitio about series. For example, which is the same as writig the series π = 3.459..., π = 3 + 0 + 4 00 + 000 + 5 0000 + 9 00000 +.... Of course this series coverges, because its sum is π! We ca t write dow a formula for the terms of this series, because the digits of π are highly irregular, but it is a coverget series oetheless. If the decimal is repeatig, the we ca write dow the series explicitly ad it will be a geometric series.
4 EVAN WARNER Example: calculate S = 9 = 00. We ca solve this usig the geometric series formula, gettig, or we ca do the followig: i decimal otatio, the umber is S = 0.09. How ca we recover a fractio from a repeatig decimal? (Last time we just recogized 0.3 as oe third because it is so commo.) I geeral, here s a trick: otice that S 00 = 0.0009, so S S 00 = 0.09 = 9 00 exactly (all other digits cacel). Solvig for S, we have 99 00 S = 9 00 = S = 9 99 =. I fact i geeral a repeatig decimal with repeatig digits is equal to the fractio with those digits as umerator ad 0 i the deomiator. For example, 0.6 = 6 53 99 ad 0.53 = 999. Two last facts about decimals. Oe, there is of course othig special about decimals, as opposed to expasios i ay other base (biary, terary, etc.); they are all still series i disguise. Two, decimal expasios are slightly ouique, which causes a lot of cofusio whe people realize that 0.9 = if they thik that real umbers are defied as decimals. They re ot! Decimals are merely a (sometimes) coveiet way of writig dow or approximatig real umbers. I fact 0.9 = is essetially the oly sort of o-uiqueess you get i decimal expasios, ad it is etirely harmless... Telescopig series. Here s aother rare form of series that we ca calculate exactly. Cosider ( ). + = This is certaily ot a geometric series, so to see what is goig o we have o choice but to start writig dow partial sums. We get ( S = ), ( S = ) ( + ), 3 ( S 3 = ) ( + ) ( + 3 3 ), 4 ad so o. But otice that if we regroup the terms, the terms cacel, ad the 3 terms cacel, ad clearly this will cotiue as we take more ad more terms. I fact for each partial sum the middle terms all cacel, leavig exactly S N = N +.
MATH SECTION NOTES 5 By defiitio, the sum is the limit of the partial sums, which is ( lim ) =. N N + So ot oly does this series coverge, it has a particularly simple sum! As aother example, cosider ( ). + = Here (check!) all of the terms i each partial sum except four cacel, ad we have S N = + N + N +. Takig the limit as N, the sum is therefore 3. Telescopig sums are rare, so you probably wo t meet ay i the wild..3. A coceptual questio. True or false: If a coverges, the =0 lim a = 0. (I other words, if a series coverges, the its terms must go to zero.) This is true; the heuristic thikig should be somethig like if the partial sums are covergig to a particular value, the the terms with very large idex had better ot be very big. Ad the proof is ot so difficult: we write each term as a differece of partial sums as follows: a = S S (why is this true?). The takig the limit as, we fid that lim S = S (because the series coverges) ad lim S = S (because this is the same sequece, just idexed slightly differetly). So lim a = S S = 0, as desired. As we will lear shortly, the coverse is very much false! That is, there are plety of sequeces a whose terms go to zero but such that =0 a does ot coverge. The simplest example is give by a = (the harmoic series). 3. April 3.. The tests we kow so far. Here are some of the tests we kow so far: Divergece test: if the limit of the terms is ot zero, the series diverges. Itegral test: a (oegative, mootoically decreasig) series coverges if ad oly if the obvious correspodig itegral does. Geometric series test: if the series is geometric, the it coverges if ad oly if the ratio of terms is less tha i absolute value (ad we ca eve calculate the sum!). p-test: if the series is of the form =N, it coverges if ad oly if p >. p Compariso test: if a series with oegative terms is bouded termwise by a coverget series, it is coverget. Similarly, if a series is greater termwise tha a diverget series with oegative terms, it is diverget.
6 EVAN WARNER Limit compariso test. Sice the limit compariso test is so ew, let s go over it agai. It states that if a ad b are all oegative terms ad a 0 < lim < b (so i particular the limit i questio exists), the a ad b either both coverge or both diverge. The limit eed ot be, although it ofte will be (ad you ca always arrage it to be such by multiplyig oe of the series by a costat, of course); the importat thig is that it exists ad is ot zero. We ted to use this test if there is a series that looks like oe we kow how to deal with except for a smaller extra term. For example, take + log. = We kow that the harmoic series = diverges, ad we hope that the extra logarithm is irrelevat. So we apply the limit covergece test to a = +log ad b =. We have a lim = lim b +log = lim + log = lim + log =. So the limit compariso test applies, ad sice we already kow that b diverges, we coclude that a does as well. Note that we could ot have used the compariso test i ay straightforward way here, because a < b for all so kowig that b diverges is ot helpful. Notice that the two compariso tests do t tell you aythig uless you already kow the covergece of aother series; we use them to bootstrap from kowledge about a simple series to kowledge about a more complicated oe. 3.. Examples. These examples alterate betwee easy ad hard. =. This series coverges by the p-test because π >. π = log. We use the itegral test because x log x has a elemetary atiderivative, usig the substitutio u = log x: dx du x log x = = log u = log log x, u so dx x log x = [log log x] x= =. The itegral does ot coverge, so the sum does ot coverge either. =. We have > for each, ad the series diverges, so by the compariso test so does this oe. = si ( ). Here we should otice first that we re evaluatig sie o values that grow closer ad closer to zero, so the covergece of the sum should oly deped o what the sie fuctio does very close to zero. Ad i fact si x < x for every oegative x, so si <, ad by the compariso
MATH SECTION NOTES 7 test the series coverges. Alteratively, we could use the limit compariso test with si x if we remember that lim x 0 x =, so certaily si lim =, ad the limit compariso test applies. = log. Here we ca just use the straightforward compariso test with the harmoic series ; certaily termwise the give sum domiates the harmoic series so because the harmoic series diverges so does this oe. 4. April 9 4.. Ratio test. The oly ew tool we have for determiig covergece of series is the ratio test, which is useful whe you see terms like k! or k because the the ratios a+ a will be particularly simple. The rule itself is simple: calculate L = lim a + a. If L <, the series coverges. If L > (icludig if the limit goes to ), the series diverges. If L = or does ot exist i aother way, the ratio test does t tell you aythig. This makes a lot of sese, give what we kow about geometric series (where the ratio is costat); morally the ratio test is kid of like a limit compariso test applied to a geometric series. Example: determie the covergece of = ()!!. This sum is tailor-made for the ratio test. We calculate a + a = (+)! + (+)! ()!! ( + )!! = + ( + )! ()! = ( + )( + ) ( + ) Clearly the limit is ifiite, so by the ratio test this series diverges. Bad example: determie the covergece of usig the ratio test. We get a + a = = (+) = ( + ). = +. The limit as is, which is uhelpful: the ratio test does t tell us aythig. I fact the ratio test will fail for ay series whose th term is a ratio of polyomials i i precisely the same way. Fortuately, we kow that this series coverges by other methods (p-test, itegral test).
8 EVAN WARNER 4.. Some exercises. What test should I use to determie the covergece of the followig series? = (log ). 4 + = = = (3+)! 4 + (3+)! log() log() Aswers, respectively: the itegral test (sice we have a u-substitutio we ca use!), the ratio test, either the ratio test directly or the limit compariso test with the previous example, ad direct compariso. To see the last oe, the hardest part is the followig algebraic maipulatio: log() log() = [e log(log )] log() [ = e log()] log(log ) = log(log ). Now you ca compare the terms of the sum to, say,. As a easy-ish exercise for yourself, carry out the tests as idicated above to see whether the series coverge or ot! 5. April 6 5.. Exam recap. The majority of both sectios was spet goig over the trickier problems from the exam, so see the exam solutios for ay ligerig questios. 5.. Alteratig series. Alteratig series are quite easy; you should be happy if you see oe! For our purposes, a alteratig series is a series of the form ( ) a, =A where a 0 for all. There is a quite geeral ad (i practice) easy to apply covergece test for such series: they coverge if () lim a = 0 ad () the sequece of terms {a } is evetually mootoically decreasig. We could state (ad prove) more geeral results for more iterestig sig patters tha + +..., if we wated, but we wo t. Example: does ( ) = coverge? Yes! Its terms, whe we igore the 0.000 ( ), go to zero (albeit rather slowly), ad are mootoically decreasig. Note that without the ( ), this series would diverge due to the p-test, so i some sese alteratigess helps us coverge. This is a geeral fact: if a coverges, the a does too. If we have a series for which the sum of the absolute value of the terms coverges, we call it absolutely coverget; if a series is coverget but ot absolutely coverget (like the example above) we call it coditioally coverget (as i, its covergece is coditioal o the fact that it has both positive ad egative terms). So istead of just havig two possibilities for the covergece of a series (covergece or divergece) we ow have three (absolute covergece, coditioal covergece, or divergece). This is actually sometimes importat: there are theorems that hold for absolutely coverget series that do ot hold for coditioally coverget series. As a example, ay rearragemet of the terms of a absolutely coverget series yields the same sum (as oe would hope!) but this dramatically fails to be true for coditioally coverget series.
MATH SECTION NOTES 9 As a postscript, a word about coditio () i the alteratig series test. I basically all examples that ayoe is goig to throw at you, it will be easily satisfied, but it is actually ecessary. Cosider the series ( ) a, where a = / if is eve ad a = / if is odd. We have lim a = 0, but the sequece is t evetually mootoic (it bouces up ad dow forever as it approaches zero). Ad i fact I ve cooked this series up i such a way that it does ot coverge: every eve term cotributes a lot i the positive directio ad every odd term cotributes ot so much i the egative directio, ad this systematic bias leads to divergece (as a good exercise, prove this!). The mootoicity coditio () rules out this sort of bias.