References: Zeta(2)=Pi^2/6,Proofs
- From: "Alex.Lupas" <alex.lupas@xxxxxxxxx>
- Date: 14 May 2007 00:36:35 -0700
Here is list (incomplete) of references regarding proofs of
the Euler's result 1+1/2^2+1/3^2+1/4^2+... =Pi^2/6 .
Perhaps of interest,/Alex
============================
[1] Roger Apery,
"Irrationalite de $ \zeta(2)$ et $\zeta(3)$",
Asterisque, 61(1979)11-13.
[2] Tom M. Apostol,
" Another elementary proof of Euler's
formula for $\zeta(2n)$,"
The American Mathematical Monthly,80(1973)425-431.
[3] Tom M. Apostol
"Mathematical Analysis,"
Addison-Wesley,1974.
[4] Tom M. Apostol,
" A proof that Euler missed: Evaluating
$\zeta(2n) $ the easy way,"
Mathematical Intelligencer,5(1983)59-60.
[5] Raymond Ayoub,"Euler and the zeta function,"
American Mathematical Monthly,81(1974)1067-1085.
[6] T.J.I'a Bromwich,
"An introduction to the theory of infinite series"
Macmillan,reprinted 1965.
[7] Robin Chapman,
" Evaluating $\zeta(2) $, " (preprint)
E-Mail author: rjc@xxxxxxxxxxxxxxxxxx
The author presents 14 proofs.
[8] Boo Rim Choe ,
"An Elementary Proof of $\sum^\infty_{n=1} 1/n^2 = \pi^2/6$,"
The American Mathematical Monthly, Vol.94, No.7(1987)662-663.
[9] Paul Erdös and Underwood Dudley,
" Some remarks and problems in number theory related
to the work of Euler,"
Mathematics Magazine,56(1983)292-298.
[10] Leonhard Euler,
" De Summis Serierum Reciprocarum,"
Commentarii Academiae Scientiarum Petropolitanae,7(1734/35),
1740,pp.123-134=
Opera Omnia, 14, 73-86.
Regarding the L. EUler work: see
http://front.math.ucdavis.edu/math.HO
[size=9]and then search Euler Leonhard, or directly a translation of
the work
"On the sums of series of reciprocals" by Leonhard Euler, 9 pages,
also :
http://arxiv.org/PS_cache/math/pdf/0506/0506415.pdf
[11] Daniel P. Giesy,
"Still another elementary proof that SUM_{k=1 to k=infty}1/
k^2=pi^2/6 ,"
Mathematics Magazine,45(1972)148-149.
[12]J.W.L. Glaisher,
(?)
Messenger of Mathematics vol.33(1903)1-20.
[13]J.W.L. Glaisher,
"Summation of certain numerical series",
Messenger of Mathematics vol.42(1913)19-34.
[14] Jean-Paul Delahayes,
"Obsession de $\pi $, "
Pour LA SCIENCE,Nr.231,January 1997.
[15] Jean-Paul Delahayes,
"Certitudes sans démonstration ?
(Identifier les constantes mathématiques nécessite des tables
numériques, de bonnes idées et d'excellentes machines), "
Pour LA SCIENCE ,Nr.249 Juillet 1998.
[16] J, Derbyshire ,
"Prime Obsession: Bernhard Riemann and the
Greatest Unsolved Problem in Mathematics",
Joseph Henry Press ,2003,ISBN 0-309-08549-7
[17] J. Elstrodt,
"Mass-und Integrationstheorie,"
Vierte,Korriegierte Auflage,Springer,2005,
see p.184, where is cited [12]
[18] E. Fabry,
"Theorie des series a termes constants"
A.Hermann & Fils,Paris,1911. ( see pp.86-88,p.121)
[19]F.Goldscheider , Arch.Math.Phys.,(3)20,323-324 (1913)
[20] Mircea Ivan
"An Elementary Method for the Calculation of an
Euler Type Series"
Automat. Comput. Appl. Math., 1(1992)109-113.
[21] Mircea Ivan,
"The Chebyshev Polynomials and an Euler Type Series"
Automat.Comput. Appl. Math.,1,no.2(1992)99-102.
[22] Dan Kalman,
"Six Ways to Sum a Series, "
The College Mathematics Journal,November 1993,
Volume 24,Number 5,pp.402-421.
[23] A. A. Karatsuba, S. M. Voronin
"The Riemann Zeta-Function"
De Gruyter, 1992.
[24] Morris Kline,
"Euler and infinite series,"
Mathematics Magazine,56(1983)307-314.
[25] Konrad Knopp,
"Theorie und Anwendung der unendliche Reihen"
Grundlagen der mathematische Wissenschaften 2,
Fuenfte Auflage,Springer Verlag,1964.
[26] R.A. Kortram, "Simple proofs for
$\sum\limits_{k=1}^{\infty}\dfrac{1}{k^2}=\dfrac{\pi^2}{6}$ and
$\sin{x}=x\prod\limits_{k=1}^{\infty}\left(1-\dfrac{x^2}
{k^2\pi^2}\right),$"
Mathematics Magazine, Vol. 69, No. 2 (Apr., 1996), pp. 122-125
[27] Bill Leonard and Harris S. Shultz,
"A computer verification of
a pretty mathematical result, "
Mathematical Gazette, 72(1988)7-10.
[28] Y.Matsuoka,
"An Elementary Proof of the Formula sum ,,, ."
Amer. Math. Monthly 68(1961)486-487.
[29] Mark B. McKinzie and Curtis D. Tuckey,
"Euler's proof of $ \sum{1/k^2}=\pi^2/6 ,$"
Annual meeting of the American Mathematical Society,
San Antonio, Texas, January 1993.
[30] Ioannis Papadimitriou,
"A simple proof of the formula
$\sum_{k=1 to k=infty}1/k^2=\pi^/6,$" The American Mathematical
Monthly, 80 (1973) 424-425.
[31] Nicholas Shea,
"Summing the series 1/1^2+1/3^2 +..., "
Mathematical Spectrum, 21(1988-89)49-55.
[32] E.L. Stark,
"Another Proof of the Formula
$\sum\limits_{k=1}^{\infty} 1/k^2 = \pi^2/6 $ ,"
The American Mathematical Monthly, Vol. 76,5(1969)552--553.
[33] E.L. Stark,
"The series $\sum\limits_{k=1}^{\infty}k^{-s}\; ,\;
s=2,3,4,...$ once more,"
Mathematics Magazine, 47(1974)197-202.
[34] A.M. Yaglom and I.M. Yaglom ,
"Non-elementary Problems Solved with Elementary Methods",
(in Romanian) Ed. Tehnica, Bucharest, 1962.
.
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