1,2

Sum_{m = 1..inf } 1/m^2.

"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

Also dilogarithm(1). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 21 2004

Also Integral_{x=0..inf} x/(exp(x)-1).

For the partial sums see the fractional sequence A007406/A007407.

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.

R. Calinger, "Leonard Euler: The First St. Petersburg Years (1727-1741)," Historia Mathematica, Vol. 23, 1996, pp. 121-166.

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.

G. F. Simmons, Calculus Gems, Section B.15,B.24 pp. 270-1,323-5 McGraw Hill 1992.

A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 261.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n=1,...,20000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)

R. Chapman, Evaluating Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6

R. W. Clickery, Probability of two numbers being coprime [Broken link]

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

Math. Reference Project, The Zeta Function, Zeta(2)

Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime"

J. Perry, Prime Product Paradox

S. Plouffe, Plouffe's Inverter, Zeta(2) or Pi**2/6 to 100000 digits

S. Plouffe, Zeta(2) or Pi**2/6 to 10000 places

A. L. Robledo, PlanetMath.org, value of the Riemann zeta function at s=2

E. Sandifer, How Euler Did It, Estimating the Basel Problem

E. Sandifer, How Euler Did It, Basel Problem with Integrals

C. Tooth, Pi squared over six

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics

H. Wilf, Accelerated series for universal constants, by the WZ method

Eric Weisstein's World of Mathematics, Dilogarithm MathWorld page

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

1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

RealDigits[N[Pi^2/6, 100]][[1]]

(PARI) \p 200; Pi^2/6

(PARI) \p 200 dilog(1) \p 200 zeta(2)

(PARI) a(n)=if(n<1, 0, default(realprecision, n+2); floor(Pi^2/6*10^(n-1))%10)

(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]

Cf. A013679, A013631, A013680, 1/A059956.

Sequence in context: A029680 A021612 A110756 this_sequence A019174 A019166 A058158

Adjacent sequences: A013658 A013659 A013660 this_sequence A013662 A013663 A013664

cons,nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009