Search: id:A013661
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%I A013661
%S A013661 1,6,4,4,9,3,4,0,6,6,8,4,8,2,2,6,4,3,6,4,7,2,4,1,5,1,6,6,6,4,6,0,2,
%T A013661 5,1,8,9,2,1,8,9,4,9,9,0,1,2,0,6,7,9,8,4,3,7,7,3,5,5,5,8,2,2,9,3,7,
%U A013661 0,0,0,7,4,7,0,4,0,3,2,0,0,8,7,3,8,3,3,6,2,8,9,0,0,6,1,9,7,5,8,7,0
%N A013661 Decimal expansion of zeta(2) = Pi^2/6.
%C A013661 Sum_{m = 1..inf } 1/m^2.
%C A013661 "In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite 
               series, Sum 1/n^2. He did it by doing some rather ingenious mathematics 
               using trigonometric functions that proved the series summed to exactly 
               Pi^2/6. How can this be? ... This demonstrates one of the most startling 
               characteristics of mathematics - the interconnectedness of, seemingly, 
               unrelated ideas." - Clawson
%C A013661 Also dilogarithm(1). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Jul 21 2004
%C A013661 Also Integral_{x=0..inf} x/(exp(x)-1).
%C A013661 For the partial sums see the fractional sequence A007406/A007407.
%D A013661 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 811.
%D A013661 R. Calinger, "Leonard Euler: The First St. Petersburg Years (1727-1741),
               " Historia Mathematica, Vol. 23, 1996, pp. 121-166.
%D A013661 Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, 
               Perseus Books, 1996, p. 97.
%D A013661 W. Dunham, Euler: The Master of Us All, The Mathematical Association 
               of America, Washington, D.C., 1999, p. xxii.
%D A013661 G. F. Simmons, Calculus Gems, Section B.15,B.24 pp. 270-1,323-5 McGraw 
               Hill 1992.
%D A013661 A. Weil, Number theory: an approach through history; from Hammurapi to 
               Legendre, Birkhaeuser, Boston, 1984; see p. 261.
%D A013661 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers,
               " Revised Edition, Penguin Books, London, England, 1997, page 23.
%H A013661 Harry J. Smith, <a href="b013661.txt">Table of n, a(n) for n=1,...,20000</
               a>
%H A013661 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A013661 D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="http://arXiv.org/
               abs/math.CA/0505270">Experimental determination of Ap'ery-like identities 
               for zeta(4n+2)</a>
%H A013661 R. Chapman, <a href="http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf">Evaluating 
               Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6</a>
%H A013661 R. W. Clickery, <a href="http://www.cacr.caltech.edu/~roy/upi/coprime.html">
               Probability of two numbers being coprime</a> [Broken link]
%H A013661 L. Euler, <a href="http://arXiv.org/abs/math.HO/0506415">On the sums 
               of series of reciprocals</a>
%H A013661 L. Euler, <a href="http://www.eulerarchive.org">De summis serierum reciprocarum</
               a>, E41.
%H A013661 Math. Reference Project, <a href="http://www.mathreference.com/lc-z,zeta2.html">
               The Zeta Function, Zeta(2)</a>
%H A013661 Math. Reference Project, <a href="http://www.mathreference.com/lc-z,cop.html">
               The Zeta Function, Odds That Two Numbers Are Coprime"</a>
%H A013661 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/paradox/
               paradox.htm">Prime Product Paradox</a>
%H A013661 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               zeta2.txt">Zeta(2) or Pi**2/6 to 100000 digits</a>
%H A013661 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap96.html">Zeta(2) or Pi**2/
               6 to 10000 places</a>
%H A013661 A. L. Robledo, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               ValueOfTheRiemannZetaFunctionAtS2.html">value of the Riemann zeta 
               function at s=2</a>
%H A013661 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/
               euler/How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf">
               Estimating the Basel Problem</a>
%H A013661 E. Sandifer, How Euler Did It, <a href="http://www.maa.org/editorial/
               euler/How%20Euler%20Did%20It%2005%20Basel%20with%20integrals.pdf">
               Basel Problem with Integrals</a>
%H A013661 C. Tooth, <a href="http://www.pisquaredoversix.force9.co.uk">Pi squared 
               over six</a>
%H A013661 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RiemannZetaFunctionZeta2.html">Link to a section of The World of 
               Mathematics</a>
%H A013661 H. Wilf, <a href="http://www.dmtcs.org/volumes/abstracts/dm030406.abs.html">
               Accelerated series for universal constants, by the WZ method</a>
%H A013661 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Dilogarithm.html">Dilogarithm MathWorld page</a>
%F A013661 Limit(n-->+oo) of (1/n)*(sum(k=1, n, frac((n/k)^(1/2)))) = zeta(2) and 
               in general have limit(n-->+oo) of (1/n)*(sum(k=1, n, frac((n/k)^(1/
               m)))) = zeta(m), m >= 2. - Yalcin Aktar (aktaryalcin(AT)msn.com), 
               Jul 14 2005
%e A013661 1.6449340668482264364724151666460251892189499012067984377355582293700074704032...
%t A013661 RealDigits[N[Pi^2/6, 100]][[1]]
%o A013661 (PARI) \p 200; Pi^2/6
%o A013661 (PARI) \p 200 dilog(1) \p 200 zeta(2)
%o A013661 (PARI) a(n)=if(n<1,0,default(realprecision,n+2); floor(Pi^2/6*10^(n-1))%10)
%o A013661 (PARI) { default(realprecision, 20080); x=Pi^2/6; for (n=1, 20000, d=floor(x); 
               x=(x-d)*10; write("b013661.txt", n, " ", d)); } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Apr 29 2009]
%Y A013661 Cf. A013679, A013631, A013680, 1/A059956.
%Y A013661 Sequence in context: A029680 A021612 A110756 this_sequence A019174 A019166 
               A058158
%Y A013661 Adjacent sequences: A013658 A013659 A013660 this_sequence A013662 A013663 
               A013664
%K A013661 cons,nonn,nice
%O A013661 1,2
%A A013661 N. J. A. Sloane (njas(AT)research.att.com).
%E A013661 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009

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