MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series............................. 4.7 Euler s Costat........................... 4 4.8 * Applicatio - Stirlig s Formula *................. 7
4.4 Series with Positive ad Negative Terms With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with both positive ad egative terms are harder to deal with. Exercise Why do t we have to separately cosider series which have oly egative terms? 4.5 Alteratig Series Oe very special case is a series whose terms alterate i sig from positive to egative. That is, series of the form ( ) + a where a 0. Example ( ) + = + 3 4 + 5... is a alteratig series. All The Way Assigmet Let s = r= ( )r+ /r. Prove that s is coverget usig the followig steps. Make sure that your proof also proves the iequality k=+ ( ) k+ k.. Prove that s + s is positive. This shows that the sequece (s ) is strictly icreasig. Do t stop oce you ve proved that (s ) ad (s +) coverge. You still have to show that the whole sequece of partial sums (s ) coverges.. Now fid s +3 s + ad prove that it is egative. This shows that (s + ) is strictly decreasig. 3. Usig the fact that s < s + + = s + < s, deduce that (s ) ad (s + ) are both coverget to the same limit. 4. To show that s coverges to s, prove that s + + s s + + ad the show that s s < for both odd ad eve N. Coclude that (s ) s. Assigmet Let s = = ( ) +. Fid a value of N so that N = ( ) + s 0 6. Theorem Alteratig Series Test Suppose (a ) is o-egative, decreasig ad ull. The the alteratig series ( ) + a is coverget. Example Sice ( ) is a decreasig ull sequece of positive terms, this test tells us right away that ( ) + = + 3 4 + 5... is coverget.
Similarly, ( ) is a decreasig ull sequece of positive terms, therefore ( ) + = 4 + 9 6 + 5... is coverget. Assigmet 3 Fid a sequece (a ) which is o-egative ad decreasig but where ( ) + a is diverget ad a sequece (b ) which is o-egative ad ull but where ( ) + b is diverget. Assigmet 4 Usig the steps below (a geeralisatio of assigmet ) prove the Alteratig Series Test. Suppose that (a ) is o-egative, decreasig ad ull. Let s = r= ( )r+ a r.. Show that s + s > 0 ad that s +3 s + < 0.. Prove that s s + a + = s + s. 3. Show that the sequeces (s ) ad (s + ) are both coverget to the same limit, s say. 4. Deduce that ( ) + a is coverget. Make sure your proof icludes the iequality s k= ( )k+ a k a. Assigmet 5 The series 3! + 5! 7! + 9!... coverges to si. Explai how to use the series to calculate si to withi a error of 0 0. Assigmet 6 Usig the Alteratig Series Test where appropriate, show that each of the followig series is coverget.. ( ) + 3 +. ( ) 3. cos π +( ) 4. π (+) 3 si 4.6 Geeral Series Series with positive terms are easier because we ca attempt to prove that the partial sums (s ) coverge by exploitig the fact that (s ) is icreasig. I the geeral case, (s ) is ot mootoic. We ca still try to apply Cauchy s test for covergece, however, sice this applies to ay sequece. Assigmet 7 Show that (s ) is a Cauchy sequece meas that for ay ɛ > 0 there exists N such that k=m+ a k < ɛ wheever > m N.
Before exploitig the Cauchy test we shall give oe ew defiitio: If a is a series with positive ad egative terms, we ca form the series a, all of whose terms are o-egative. Defiitio The series a is absolutely coverget if a is coverget. Example The alteratig series ( ) + is absolutely coverget because ( ) + = is coverget. The series ( ) ( is absolutely coverget because ) coverges. Assigmet 8 Is the series ( ) + absolutely coverget? Exercise For what values of x is the Geometric Series x absolutely coverget? Absolutely coverget series are importat for the followig reaso. Theorem Every absolutely coverget series is coverget. Assigmet 9 Let s = i= a i ad t = i= a i. Prove this result usig the followig steps:. Show that s s m t t m wheever > m.. Show that if t is coverget the s is Cauchy ad hece coverget. Assigmet 0 Is the coverse of the theorem true: coverget? Every coverget series is absolutely The Absolute Covergece Theorem breathes ew life ito all the tests we developed for series with o-egative terms: if we ca show that a is coverget, usig oe of these tests, the we are guareteed that a coverges as well. Exercise 3 Show that the series si is coverget. si We see that 0. Therefore si is coverget by the Compariso Test. It follows that si is coverget by the Absolute Covergece Theorem. 3
The Ratio Test ca be modified to cope directly with series of mixed terms. Theorem Ratio Test Suppose a 0 ad a+ a l. The a coverges absolutely (ad hece coverges) if 0 l < ad diverges if l >. Proof. If 0 l <, the a coverges by the old Ratio Test. Therefore a coverges by the Absolute Covergece Theorem. If l >, we are guareteed that a diverges, but this does ot, i itself, prove that a diverges (why ot?). We have to go back ad modify our origial proof. We kow that there exists N such that a+ a l < (l ) whe > N. Therefore, < a+ ( + l) < a whe > N. It follows that 0 < a N+ < a N+ ad by iductio that 0 < a N+ < a whe > N +. Clearly the sequece ( a ) is ot ull, hece (a ) is ot ull. This beig the case, a diverges by the Null Sequece Test. Example Cosider the series x. Whe x = 0 the series is coverget. (Notice that we caot use the Ratio Test i this case.) Now let a = x a+. Whe x 0 the a = x+ + x = + x x. Therefore x is coverget whe x < ad diverget whe x >, by the Ratio Test. What if x =? Whe x = the x = which is diverget. Whe x = the x = ( )+ which is coverget. Theorem Ratio Test Extesio Suppose a 0 ad a+ a, the a diverges. Assigmet Prove this theorem. Assigmet Determie for which values of x the followig series coverge ad diverge. [Make sure you do t eglect those values for which the Ratio Test does t apply.]. x!. x 3.!x 4. (x) 5. (4x) 3 + 6. ( x) 4.7 Euler s Costat Our last aim i this booklet is to fid a explicit formula for the sum of the alteratig series: + 3 4 + 5 6 +... 4
O the way we shall meet Euler s costat, usually deoted by γ, which occurs i several places i mathematics, especially i umber theory. 5
replacemets f() f() f(3) f(4) f( ) f() 3 4 5 + Figure : Calculatig a lower boud of a itegral. Assigmet 3 Let D = i= i + x dx = i= i log( + ).. Draw a copy of figure ad mark i the areas represeted by D.. Show that (D ) is icreasig. 3. Show that (D ) is bouded - ad hece coverget. The limit of the sequece D = ad is usually deoted by γ. i= i log(+) is called Euler s Costat Assigmet 4 Show that ( ) i+ i= i = log + D D. Hece evaluate ( ) + Hit: Use the followig idetity:. Euler s Costat The limit of the sequece D = i= log( + ) i is called Euler s Costat ad is usually deoted by γ (gamma). Its value has bee computed to over 00 decimal places. Its value to 4 decimal places is 0.57756649053. No-oe kows whether γ is ratioal or irratioal. + 3 4 + + = + + 3 + + ( + 4 + ) 6 +... 6
frag replacemets a b b a 5 5 a 4 b 4 a 3 b 3 a b b 3 4 5 Figure : Approximatig the itegral by the mid poit. 4.8 * Applicatio - Stirlig s Formula * Usig the alteratig series test we ca improve the approximatios to! that we stated i workbook 4. Take a look at what we did there: we obtaied upper ad lower bouds to log(!) by usig block approximatios to the itegral of log xdx. To get a better approximatio we use the approximatio i figure. Now the area of the blocks approximates log xdx except that there are small triagular errors below the graph (marked as b, b, b 3,... ) ad small triagular errors above the graph (marked as a, a 3, a 4,... ). Note that log! = log + log 3 + + log = area of the blocks. Assigmet 5 Use the above diagram to show: log! ( + ) log + = b + a b + a 3 b 3 + b + a [Hit: log xdx = log + ] The curve log is cocave dow ad it seems reasoable, ad ca be easily proved (try for yourselves), that a b a + ad lim a = 0. Assigmet 6 Assumig that these claims are true, explai why b + a b + C as. This proves that log! = (+ ) log +Σ where Σ teds to a costat 7
as. Takig expoetials we obtai:! costat e What is the costat? This was idetified with oly a little more work by the mathematicia James Stirlig. Ideed, he proved that:! e π as a result kow as Stirlig s formula. Check Your Progress By the ed of this Workbook you should be able to: Use ad justify the Alteratig Series Test: If (a ) is a decreasig, ull sequece of o-egative terms the ( ) + a is coverget. Use the proof of Alteratig Series Test to establish error bouds. Say what it meas for a series to be absolutely coverget ad give examples of such series. Prove that a absolutely coverget series is coverget, but that the coverse is ot true. Use the modified Ratio Test to determie the covergece or divergece of series with positive ad egative terms. Prove that = ( ) + = log. 8
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