The Harmonic Series Diverges Again and Again

Transcription

1 The Harmoic Series Diverges Agai ad Agai Steve J. Kifowit Prairie State College Terra A. Stamps Prairie State College The harmoic series, = = 3 4 5, is oe of the most celebrated ifiite series of mathematics. As a couterexample, few series more clearly illustrate that the covergece of terms to zero is ot sufficiet to guaratee the covergece of a series. As a kow series, oly a hadful are used as ofte i comparisos. From a pedagogical poit of view, the harmoic series provides the istructor with a wealth of opportuities. The leaig tower of lire Johso 955 a.k.a the book stackig problem is a iterestig hads-o activity that is sure to surprise studets. Applicatios such as Gabriel s weddig cake Flero 999 ad Euler s proof of the divergece of /p p prime Duham 999, pages ca lead to some very ice discussios. Ad the proofs of divergece are as etertaiig as they are educatioal. A quick survey of moder calculus textbooks reveals that there are two very popular proofs of the divergece of the harmoic series: those fashioed after the early proof of Nicole Oresme ad those comparig k= /k ad /x dx. While these proofs are otable for their cleveress ad simplicity, there are a umber of other proofs that are equally simple ad isightful. I this article, the authors survey some of these divergece proofs. Throughout, H is used to deote the th partial sum of the harmoic series. That is, H = 3, =,, 3,.... A commo thread coectig the proofs is their accessibility to first-year calculus studets. To appear i The AMATYC Review, Sprig 006

2 The Proofs Though the proofs are preseted i o particular order, it seems fittig to begi with the classical proof of Oresme. Proof Nicole Oresme s proof dates back to about 350. While the proof seems to have disappeared util after the Middle Ages, it has certaily made up for lost time. Proof: Cosider the subsequece {H k} k=0. H = = 0, H = =, H 4 = 3 4 > 4 =, 4 H 8 = > = 3. 8 I geeral, H k k. Sice the subsequece {H k} is ubouded, the sequece {H } diverges. M, Usig the same type of argumet, oe ca show that for ay positive iteger H M k M k. M A slight variatio o this theme is preseted ext. Proof The followig proof is derived from oe give by Hosberger 976, page 98. Proof: There are 9 oe-digit umbers, to 9, whose reciprocals are greater

3 tha /0. Therefore H 9 > 9 0. There are 90 two-digit umbers, 0 to 99, whose reciprocals are greater tha /00. Therefore H 99 > =. 0 Cotiuig with this reasoig, it follows that 9 H 0k > k. 0 Sice the subsequece {H 0 k } is ubouded, the sequece {H } diverges. Proof 3 Credit for this proof goes to Pietro Megoli. His proof dates back to the middle of the 7th cetury. The presetatio give here is similar to Duham s 990, pages Proof: First otice that = > =, =, 3, 4,... so that 4 > 3, 5 7 > 6, 8 0 > 9, etc. Now suppose that the harmoic series coverges with sum S. The S = > = S. The cotradictio S > S cocludes the proof. The iequality used by Megoli, >, =, 3, 4,..., is a special case of the harmoic mea/arithmetic mea iequality: < x. x i= i 3

4 Based o this iequality, oe ca also show that for positive itegers k ad j, k k k j > j j k. From this, a umber of other proofs ca be derived. For example, cosecutive terms of the harmoic series could be grouped i such a way that the sum of each group is at least oe. This would lead to the subsequece {H 3 /} whose th term is bouded below by : H 3 /, =,, 3,.... Proof 4 This proof makes use of a iterestig example of a bouded, mootoe sequece. It is well kow that the sequece uder ivestigatio coverges to l, but its limit is ot relevat to the proof. Proof: Cosider the sequece {S } =, where Sice S = H H =.... S S = the sequece {S } is icreasig. Also, = S S = k= > 0, k k= = <, ad therefore the sequece {S } is positive ad bouded above. It follows that {S } coverges to a positive umber betwee / ad. Sice S = H H, the sequece {H } must diverge. Notice that the fact that S / is eough to show that the sequece of partial sums {H } is ot a Cauchy sequece ad is therefore diverget. Proof 5 Hosberger 976, page 78 gives this proof as a solutio of oe of his exercises. I additio to a familiar expoet law, the proof makes use of the iequality 4

5 e x > x, which holds for ay ozero x. Proof: Cosider the sequece {e H } =. e H = exp 3 4 = e e / e /3 e /4 e / > 3 4 = =. Sice {e H } is ubouded, {H } is ubouded. Proof 6 The ext proof also came from Hosberger 976, page 0, but credit was give to Leoard Gillma. Proof: Suppose that the harmoic series coverges with sum S. The S = = > = S. The cotradictio S > S cocludes the proof. Notice that a whole family of similar proofs ca be derived by cosiderig terms i groups of three, four, etc. 5

6 Proof 7 The followig proof was give by Cusumao 998. Proof: Suppose that the harmoic series coverges with sum S. The S = = = It follows that This cotradictio cocludes the proof. S = S This proof is very closely related to the previous proof. I fact, the terms o opposite sides of the iequality of Proof 6 differ by /, /, /30, /56, etc. Just as Gillma s proof has variatios, which are based o groupig larger collectios of terms, so there are variatios o Cusumao s. For example, after placig terms ito groups of three, oe would fid that 5 S = S Proof 8 This ext proof was first preseted by Cohe ad Kight 979 ad later by Ecker 997. Proof: Suppose the harmoic series coverges with sum S. The 4 = S. Therefore the sum of the odd-umbered terms, 3, must be the other half of S. However this is impossible sice > for each positive iteger. This cotradictio cocludes the proof. 6

7 Proof 9 This proof without words compares H ad /x dx. Its variatios, icludig those ivolvig the itegral test, are amog the most popular proofs of the divergece of the harmoic series. Proof: fx = x dx x = l < 3 Proof 0 While ot completely rigorous, this proof is thought-provokig oetheless. It may provide a good exercise for studets to fid possible flaws i the argumet. Proof: 3 = x x x dx = = =. x dx k k=0 x dx 7

8 Proof The proof above is very similar to oe give by Euler i 748 i his Itroductio i aalysi ifiitorum. Euler first established a series represetatio for l x: l x = x x x3 3 x4 4 x5 5. Havig doe so, his proof was simple, though it hardly meets today s stadards of rigor. Proof: Start by writig l x as a power series: l x = x x x3 3 x4 4 x5 5. It follows that l 0 =. Proof The followig proof was give by Jacob Beroulli i his 689 Tractatus de seriebus ifiitis. The presetatio here is similar to that give by Duham 999, page 30. Proof: First otice that if c is a iteger ad c >, the c c c c c c = c. Now add /c to the left-had ad right-had sides to establish that It follows that = = c c c c

9 Proof 3 Although Jacob Beroulli also gave this ext proof, he credited its discovery to his brother Joha. A ejoyable accout of the history of the proof ca be foud i the works of Duham 987, 990. We preset a moder versio of the Beroulli proof. Proof: Cosider the series = = = =. As it is writte o the right, the series is telescopig ad coverges to. With this series servig as a illustratio, ote that =k = =k =, k =,, 3,.... k Now suppose that the harmoic series coverges with sum S. The S = = = = = = 3 = S. = 6 0 =3 The cotradictio S = S cocludes the proof. Proof 4 Here is a uusual proof by cotradictio. I this proof, we examie the subsequece {H T }, where T = / is the th triagular umber. 9

10 Proof: Suppose that the harmoic series coverges with sum S. The S must be greater tha sice H 4 = 5/. Now otice that S = > = = = = S. The iequality S > S implies S <. This cotradictio cocludes the proof. Proof 5 Several of the proofs i this article have bee based o a commo theme: if the sequece {σ } grows fast eough, the the correspodig subsequece of the harmoic series {H σ } is bouded below by a liear fuctio of. This theme makes aother appearace i the followig proof. Proof: First otice that if k is a iteger ad k >, the k! k! k! k! > = k! k! k. Now cosider the subsequece {H! }. H! =! k= > = k = k= k= = H. k= k k k! k! 0

11 It follows that H! > H! H >. Therefore {H! } is ubouded, ad the harmoic series diverges. More or Less Proofs The divergece of the harmoic series is sometimes proved by appealig to more geeral results. The most commo examples of this ivolve the itegral test or the p-series test. I this sectio, we preset a umber of ofte overlooked results, each of which immediately implies the divergece of the harmoic series. Proof 6 The followig modified th term test quickly shows that the harmoic series caot coverge. Suppose {a } is both positive ad decreasig. If a coverges, the lim a = 0. Goar 999 describes a episode i the history of aalysis i which Louis Olivier used the coverse of this result as a covergece test. Olivier s mistake was corrected by 6-year-old Neils Abel, shortly before Abel s utimely death. Proof 7 After learig the p-series test, studets ofte believe that the harmoic series forms a kid of boudary betwee the coverget ad diverget series. Fortuately, it is easy to fid couterexamples. The ext result shows that a diverget series ca actually be very much less term-by-term tha the harmoic series. A simple proof of this result was give by Ash 997. The divergece of the harmoic series follows by settig d = for each. Suppose = d is a diverget series with positive terms. If s = d d... d, the = d /s diverges. Proof 8 Eve though ifiite products are rarely discussed i first-year calculus, it is atural for studets to questio their existece ad covergece. This ext

12 result could easily be icluded i a short itroductio to the topic. It is similar to Proof 5, ad it shows that products ad sums share at least oe very importat property. Suppose {a } is a sequece with oegative terms. The a ad a either both coverge or both diverge. Proof 9 The ext result has bee proposed as a alterative to the itegral test Jugck 983. Let f be positive ad oicreasig o [k,, k a positive iteger, ad let g be ay atiderivative of f. The =k f coverges if ad oly if g is bouded above o [k,. The proof of this result provides a iterestig applicatio of the mea value theorem. Note that the itegral test follows immediately by lettig gx = x ft dt. k Proof 0 While this result is the least elemetary of our collectio, we iclude it because of its relative obscurity ad its possible appeal to studets. The test is due to Abu-Mostafa 984, ad it has a very ice geometric iterpretatio. The divergece of the harmoic series follows by settig fx = x. Let f be a real fuctio such that d f/dx exists at x = 0. The = f/ coverges absolutely if ad oly if f0 = f 0 = 0. Coclusio The harmoic series has diverged agai ad agai for well over six ceturies. I this article, we surveyed a umber of the divergece proofs, but there are others. We chose to highlight those that are particularly accessible to first-year calculus studets. I our experiece, these proofs have excited ad motivated studets. Some proofs have provided historical cotexts, ad some have provided coectios to other topics. I ay case, as we see it, there are at least H good reasos to share these proofs with your studets.

13 Refereces Abu-Mostafa, Y. S A differetiatio test for absolute covergece. Mathematics Magazie 574, 8 3. Ash, J. M Neither a worst coverget series or a best diverget series exists. College Mathematics Joural 84, Cohe, T. ad W. J. Kight 979. Covergece ad divergece of = /p. Mathematics Magazie 53, 78. Cusumao, A The harmoic series diverges. America Mathematical Mothly 057, 608. Duham, W The Beroullis ad the harmoic series. College Mathematics Joural 8, 8 3. Duham, W Jourey Through Geius: The Great Theorems of Mathematics. Joh Wiley ad Sos. Duham, W Euler: The Master of Us All. The Mathematical Associatio of America. Ecker, M. W Divergece of the harmoic series by rearragemet. College Mathematics Joural 83, Flero, J. F Gabriel s weddig cake. College Mathematics Joural 30, Goar, M Olivier ad Abel o series covergece: A episode from early 9th cetury aalysis. Mathematics Magazie 75, Hosberger, R Mathematical Gems II. The Mathematical Associatio of America. Johso, P. B Leaig tower of lire. America Joural of Physics 34, 40. Jugck, G A alterative to the itegral test. Mathematics Magazie 564,