Sum of the generalized harmonic series with even natural exponent

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Herbert Heath
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1 Rendiconti di Matematica, Serie VII Volume 33, Roma (23), 9 26 Sum of te generalized armonic erie wit even natural exponent STEFANO PATRÌ Abtract: In ti paper we deal wit real armonic erie, witout conidering teir complex extenion to te Riemann zeta function. It i well known tat te generalized armonic erie are convergent if te exponent i greater tan one, wile tey are divergent if te exponent i one or le tan one. Furter, if te exponent i an even natural number, tere exit te um of te erie in cloed form being equal to π time a rational number. Ti um wa calculated for te firt time by Euler (ee, for example, [2]) troug Taylor expanion of te function in x/x and ten by Fourier troug te expanion of uitable periodic function. Furter, te formula /n2 = π 2 /6 can be proved by uing Caucy Reidue Calculu or Weiertraß Product Teorem (ee, for example, te firt five book in te reference of []). In recent time te formula /n2 = π 2 /6 a been proved in many oter way (ee [, 3, 4, 5, 6]) troug elementary goniometric argument or imple propertie of te erie and product expanion. Many of tee metod, owever, apply only to te cae /n2. In ti paper we obtain te um of all generalized armonic erie wit an even natural exponent by calculating te eigenvalue of te differential operator derivative of order defined on a certain Hilbert pace and ten by inverting uc operator, in order to obtain te um of te erie a trace of te invere operator. General Cae To calculate te um of te generalized armonic erie / ( n ), let u conider, for eac fixed poitive k N, te linear differential operator T := d Key Word and Prae: Eigenvalue Trace of operator Green function A.M.S. Claification: 34B9, 34B27.

2 2 STEFANO PATRÌ [2] defined on te et D = u L 2 [,]: d2i u() dx 2i Since in ti pace te general eigenvalue equation } = d2i u() dx 2i =, i =,, 2,...,k C (,). T ψ n (x) =λ n ψ n (x) (.) aume te form d [ ( nπ C in x = ( )k n π [ ( nπ C in x, we recognize tat te eigenvalue λ n and te eigenfunction ψ n (x) of te linear operator T are λ n = ( )k n π ( nπ ) and ψ n (x) =C in x repectively. We oberve tat equation (.) defined on te et D can be een, in te frame of quantum mecanic, a a generalization of te time independent Scrödinger equation aociated to a free particle on te interval [,]. Te invere operator of a differential operator i an integral one and by virtue of te propertie of te invere operator (in our cae T ), we ave λ n = ( ) k n π =Tr(T )=Tr [ ( d ) ] (.2) were Tr indicate te trace of te operator. To invert an operator T, for fixed k, weavetofindtekernel G(x, ) uc tat [ ] d G(x, ) u(x) dx = u() (.3) dx were G(x, ) i te Green function of te operator T. By iterating an integration by part and uing te condition on te derivative of even order for te function u D, te left-and ide of te equation (.3)

3 [3] Sum of ome armonic erie 2 become [ ] d G(x, ) u(x) dx dx = G(x, ) d u(x) dg(x, ) dx + d2 G(x, ) d 3 u(x) dx 2 3 G(x, ) + u(x) dx d3 G(x, ) dx 3 = G(x, ) d u(x) + d2 G(x, ) dx 2 G(x, ) + u(x) dx. d 2 u(x) 2 d 4 u(x) 4 d 3 u(x) 3 + d G(x, ) + + d 2 G(x, ) 2 By impoing on te Green function te boundary condition d 2i G(x, ) dx 2i du(x) dx u(x) = d2i G(x, ) = (.4) x= x= dx 2i for all i =,, 2,...,k, te left-and ide of te equation (.3) become [ ] d G(x, ) u(x) dx = dx By ubtituting te (.5) into te (.3), we obtain from wic te relation [ d G(x, ) ] u(x) dx = u() = follow, were δ(x ) i te Dirac δ-function. Te olution of (.7) i G(x, ) = G(x, ) u(x) dx. (.5) δ(x ) u(x) dx (.6) d G(x, ) = δ(x ) (.7) G (x, ) if x [,] G + (x, ) if x [, ]

4 22 STEFANO PATRÌ [4] were G (x, ) and G + (x, ) are two polynomial of degree wit repect to te variable x, tat i G (x, ) =P (x) and G + (x, ) =Q (x). By integrating (.7) wit ɛ >, we obtain te equation +ɛ ɛ d G(x, ) +ɛ dx = ɛ δ(x ) dx from wic, in te limit ɛ +,tejump dicontinuity condition d Q (x) d P (x) = (.8) follow. By ubtituting te olution G(x, ) of te equation (.7) into te left-and ide of te (.6), we obtain P (x) u(x) dx + Q (x) u(x) dx = u(). (.9) By iterating an integration by part and uing te condition (.4) wit te propertie of te et D, te equation (.9) become [ [P (x) Q (x d u(x) dp (x) dq ] (x) d 2 u(x) dx dx 2 [ d 2 ] P (x) + dx 2 d2 Q (x) d 3 u(x) dx u() =u() were te lat term of te left-and ide a coefficient by virtue of (.8). Te kernel G(x, ) of te invere operator given in te (.3) i ten G(x, ) = P (x) if x [,] Q (x) if x [, ] (.) were te 4k parameter ( parameter for eac polynomial) are obtained a olution of te algebraic linear ytem coniting of te 4k linear equation repreenting te boundary condition (.4) in and, te continuity in x = of te derivative up to te order 2 for te (.) to be an identity and te jump dicontinuity in x = of te derivative of order.

5 [5] Sum of ome armonic erie 23 Ti linear ytem i ten of te form d 2i P (x) dx 2i d j P (x) dx j d Q (x) = d2i Q (x) dx 2i = = dj Q (x) dx j d P (x) = (.a) (.b) (.c) for all i =,, 2,...,k and for all j =,, 2,..., 2. At ti point, in analogy wit te cae of finite dimenion in wic te trace of an endomorpim A =(a ρσ ) i given by te um of it diagonal element, tat i Tr(A) = σ a σσ, tetrace of te invere operator ( d /) i ten Tr [ ( d ) ] = G(, ) d. (.2) By ubtituting te trace obtained in te rigt-and ide of (.2) into te equation (.2) and implifying te factor ( ) k, π,, we finally obtain te um of te generalized armonic erie n. 2 Firt example: cae of exponent m =2 Let u conider te linear differential operator T 2 := d2 dx 2 and it eigenvalue equation aving te form wit te eigenvalue It ten follow d 2 [ ( nπ dx 2 C in x = n2 π 2 [ ( nπ 2 C in x, n 2 λ n = n2 π 2 2. = π2 2 Tr [ ( d 2 dx 2 ) ]}. (2.)

6 24 STEFANO PATRÌ [6] In order to invert te operator T 2, we ave to determine it Green function ax + b if x [,] G(x, ) = a x + b if x [, ] and ave to olve te correponding algebraic linear ytem given by te equation (.a), (.b) and (.c), woe form for ti cae i b = a + b = a + b = a + b a a =. Te olution of ti algebraic linear ytem i Ten we ave a =, b =, a =, b =. G(x, ) = ( )x x if x [,] if x [, ]. According to (.2), te trace of te invere operator (T 2 ) i ten [ Tr (T 2 ) ] ( ) 2 = G(, ) d = d = 2 6. (2.2) By ubtituting (2.2) into (2.), we finally obtain te well-known reult n 2 = π Second example: cae of exponent m =4 Let u conider te linear differential operator T 4 := d4 dx 4 and it eigenvalue equation aving te form d 4 [ ( nπ dx 4 C in x = n4 π 4 [ ( nπ 4 C in x,

7 [7] Sum of ome armonic erie 25 wit te eigenvalue It ten follow n 4 = π4 4 λ n = n4 π 4 4. Tr [ ( d 4 dx 4 ) ]}. (3.) In order to invert te operator T 4, we ave to determine it Green function ax G(x, ) = 3 + bx 2 + cx + d x [,] a x 3 + b x 2 + c x + d x [, ] and ave to olve te correponding algebraic linear ytem given by te equation (.a), (.b) and (.c), woe form for ti cae i d = a 3 + b 2 + c + d = b = 6a +2b = a 3 + b 2 + c + d = a 3 + b 2 + c + d 3a 2 +2b + c =3a 2 +2b + c 6a +2b =6a +2b 6a 6a =. Te olution of ti algebraic linear ytem i a = 6 ( ), b =, c = , d = a = 6, Ten we ave 6 G(x, ) = b = 2, c = , d = 3 6. ( ) ( x ) 3 2 x if x [,] 2 x 3 ( 6 x ) x Te trace of te invere operator (T 4 ) i ten [ ( ) d 4 ] ( ) 4 Tr dx 4 = if x [, ]. d = 4 9. (3.2)

8 26 STEFANO PATRÌ [8] By ubtituting (3.2) into (3.), we finally obtain te well-known reult n 4 = π4 9 REFERENCES [] Boo Rim Coe: An Elementary Proof of k= /k2 = π 2 /6, Te American Matematical Montly, (7) 94 (987), [2] P. Eymard J. P. Lafon: Te Number π, American Matematical Society, 24. [3] D. P. Giey: Still Anoter Elementary Proof Tat k= /k2 = π 2 /6, Matematic Magazine, (3) 45 (972), [4] Y. Matuoka: An Elementary Proof of te Formula k= /k2 = π 2 /6, Te American Matematical Montly, (5) 68 (96), [5] I. Papadimitriou: A Simple Proof of te Formula k= /k2 = π 2 /6, Te American Matematical Montly, (4) 8 (973), [6] E. L. Stark: Proof of te Formula k= /k2 = π 2 /6, Te American Matematical Montly, (5) 76 (969), Lavoro pervenuto alla redazione il 8 maggio 22 ed accettato per la pubblicazione il 2 ettembre 22 Indirizzo dell Autore: Stefano Patrì Metod and Model for Economic Territory and Finance Sapienza Univerity of Rome Italy addre: tefano.patri@uniroma.it

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te