The Basel Problem - Numerous Proofs

1 IntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesThe Basel ProblemNumerous ProofsBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarApril 11, 2013 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesAbstractTheBasel Problemwas first posed in 1644 and remainedopen for 90 years, until Euler made his first waves in themathematical community by solving it. During his life, hewould present three different solutions to the Problem ,which asks for an evaluation of the infinite series k= then, people have continually looked for new,interesting, and enlightening approaches to this sameproblem. Here, we present 5 different solutions, drawingfrom such diverse areas as complex analysis, calculus,probability, and Hilbert space theory. Along the way, we llgive some indication of the Problem s intrigue W.

2 SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 References0 IntroductionHistoryIntrigue1 Proof:sinxand L H pital2 Proof:sinxand MaclaurinAnalysis:sinxas aninfinite product3 Proof: Integral on[0,1]24 Proof:L2[0,1]and Parseval5 Proof: Probability Densities6 ReferencesBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesHistoryPietro MengoliItalian mathematician and clergyman (1626 1686) [5]PhDs in math and civil/canon lawAssumed math chair at Bologna after adviser Cavalieri diedPriest in the parish of Santa Maria Maddelena in BolognaKnown (nowadays) for work in infinite series:Proved: harmonic series diverges, alternating harmonicseries sums toln 2, Wallis product for is correctDeveloped many results in limits and sums that laidgroundwork for Newton/LeibnizWrote in abstruse Latin ; Leibniz was influenced by him [6]Brendan W.

3 SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesHistoryProblem Statement & Early WorkMengoli posed theBasel Problemin the numerical value of:1 +14+19+116+125+ = k=11k2 Wallis, 1655: I know it to 3 decimal places. Jakob Bernoulli, 1689: It s less than 2. Help us out! Johann/Daniel Bernoulli, 1721: It s about85. Goldbach, 1721ish: It s between4135= Leibniz, DeMoivre: .. ??? Part of the difficulty is that the series convergesslowly:n= 102 1place,n= 103 2places,n= 105 4places,n= 106 5placesBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesIntrigueEuler Emerges!Since this had stumped so many brilliant minds, Euler ssolution in 1735 (at age 28) brought him immediate was born in Basel . ( Problem name comes frompublishing location of Jakob Bernoulli sTractatus deseriebus infinitis, though.)

4 Studied under Johann Bernoulli, starting working on it by 1728, calculating partial a more rigorous proof in 1741, and a third in 1755 His techniques inspired Weierstrass (to rigorize his methodsand develop analysis) and Riemann (to develop the zetafunction and the Prime Number Theorem) in the 1800sBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesIntrigueRiemmann function: definition (s) = n=11ns=11s+12s+13s+ for<(s)>1 Non-series definition: (s) = 2s s 1sin( s2) (1 s) (1 s)where (z) = 0tz 1e tdtis the analytic extension of the factorial function k N. ( 2k) = 0; these hypothesis:<z=12for every nontrivial W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesIntrigueRiemmann function: applicationsIf we knew (s), we could evaluatef(x) = Li(x) Li(x ) log(2) + xdtt(t2 1) log(t)whereLi(x) = 0dxlog(x)and is overnontrivial rootsof (s).

5 This would help us find the primes function (x)by (x) =f(x) 12f(x1/2) 13f(x1/3) Riemann was likely motivated byPrime Number Theorem:limn (n)n/logn= 1 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesIntrigueEuler s work on Euler s 1741 proof actually adapted his earlier method to find k N. (2k) =( 1)k 1(2 )2k2(2k)!B2kwhereB2kis theBernoulli Number, defined byBm= 1 m 1 k=0(mk)Bkm k+ 1withB0= 1andB1= , not much is known about (2k+ 1).Closed form for (3)is open! (It s irrational [1], proven 1979.)Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: HistoryL H pital publishedAnalyse des infiniment petitsin calc text for many years, until ..Euler publishedInstitutiones calculi differentialisin standard!

6 [7]Euler s book includes a discussion of indeterminate gave no credit to L H pital (who, in turn, gave no creditto Johann Bernoulli), but he did state and prove the rule,and provide several striking examples, includinglimx 1xx x1 x+ lnx= 2andn k=1k=n(n+ 1)2and ..Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Sketch1. Writesinxas an infinite product of linear factors(More on this later .. )2. Takelogof both sides to get a sum3. Differentiate4. Make a surprising change of variables5. Plug inx= 0and use L H pital thrice6. Sit back and smile smugly at your brillianceBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Detailssint=t(1 t )(1 +t )(1 t2 )(1 +t2 ) sincesinthas roots precisely att Zsin( y) = y(1 y) (1 +y)(2 y2)(2 +y2) = y(1 y2)(4 y24)(9 y29) ln(sin( y)) = ln + lny+ ln(1 y2)+ ln(4 y2) ln 4 + Differentiate with respect toy.

7 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Detailsln(sin( y)) = ln + lny+ ln(1 y2)+ ln(4 y2) ln 4 + cos( y)sin( y)=1y 2y1 y2 2y4 y2 2y9 y2 1y+11 y2+14 y2+19 y2+ =12y2 cos( y)2ysin( y)COV:y= ix y2= x211 +x2+14 +x2+19 +x2+ = 12x2+ cos( i x)2ixsin( i x)Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: DetailsUseEuler s Formula(he s everywhere!) to find:cos(z)sin(z)=12(eiz+e iz)12i(eiz e iz)=i(e2iz+ 1)e2iz 1 cos( i x)2ixsin( i x)= 2ix i(e2 x+ 1)e2 x 1= 2x (e2 x 1)+ 2e2 x 1= 2x+ x(e2 x 1)Substitute this back into the previous equation ..Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Details11 +x2+14 +x2+19 +x2+ = 12x2+ 2x+ x(e2 x 1) cos( i x)2ixsin( i x)= x 12x2+ x(e2 x 1)= xe2 x e2 x+ x+ 12x2e2 x 2x2 Plug inx= 0:LHS is the desired sum, k= is00.

8 L H pital to the rescue!Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Details xe2 x e2 x+ x+ 12x2e2 x 2x27 e2 x+ 2 2xe2 x4xe2 x+ 4 x2e2 x 4x7 3xe2 xe2 x+ 4 xe2 x+ 2 2x2e2 x 17 34 + 4 2x+ 2 e 2 x7 34 + 2 7 26 Everything works!Because Euler said so. And it W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 1: Summary1. Writesinxas an infinite product of linear factors2. Takelogof both sides to get a sum3. Differentiate4. Make a surprising change of variables5. Plug inx= 0and use L H pital Cite Euler What a marvelous derivation it was. It boasted an all-star cast oftranscendental functions: sines, cosines, logs, and exponentials. It rangedfrom the real to the complex and back again. It featured L H pital s Rule ina starring role.

9 Of course, none of this would have happened without thefluid imagination of Leonhard Euler, symbol manipulator extraordinaire. William Dunham [7]Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: History & SketchEuler s (and the world s) first proof, from of the easiest methods to Writesinxas an infinite product of linear factors(More on this later .. )2. Also find the Maclaurin series forsinx3. Compare the coefficients ofx34. Marvel at the coincidenceBrendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: Detailssin( x) = x(1 x2)(1 x24)(1 x29) = x+ x3[1 +14+19+116+ ]+ x5[11 4+11 9+ +14 9+ ]+ sin( x) = x ( x)33!+( x)55! = x 36x3+ 5120x5 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: Detailssin( x) = x(1 x2)(1 x24)(1 x29) = x x3[1 +14+19+116+ ]+ x5[11 4+11 9+ +14 9+ ] sin( x) = x ( x)33!

10 +( x)55! = x 36x3+ 5120x5 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: DetailsThus, [1 +14+19+116+ ]= 36 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: DetailsThus,1 +14+19+116+ = 26 Brendan W. SullivanCarnegie Mellon UniversityMath Grad Student SeminarThe Basel ProblemIntroPf 1Pf 2Pf 3Pf 4Pf 5 ReferencesProof 2: Summary1. Writesinxas an infinite product of linear factors2. Also find the Maclaurin series forsinx3. Compare the coefficients ofx3 Not particularlychallenging, so to speak. You could present thisto a Calc II class and be convincing!Hints at the deeper relationships between actual validity depends heavily on complex analysis, andwould only be officially resolved in the mid 1800s by Question:Why can we factor sinxinto linear terms?