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# POLYNOMES DE TYPE LEGENDRE ET APPROXIMATIONS DE LA CONSTANTE D'EULER

11 pages

• cours - matière potentielle : route

POLYNOMES DE TYPE LEGENDRE ET APPROXIMATIONS DE LA CONSTANTE D'EULER T. RIVOAL Resume. We propose a simple method to accelerate significatively the convergence of Sn = ∑n?1 j=1 1/j ? log(n) towards Euler's constant ? : we construct linear combinations of consecutive values of Sn, to which are assigned certain binomial weights, obtained thanks to a classical integral representation of ??Sn and certain special polynomials, of Legendre type. As a special case, we recover an approximation of ? due to Elsner, obtained by a different method. Our approach also applies to the number log(4/pi) which, as Sondow has noted, is in a sense an alternating analogue of ? ; this enables us to produce an apparently new expression of pi/4 as an infinite product, which can be viewed as an analogue of Vacca's series for ?. Finally, although this method cannot prove the irrationality of ?, it is similar to the one used by Alladi and Robinson to prove the irrationality of log(2) by means of Legendre polynomials. 1. Introduction La constante d'Euler ? est definie comme la limite, lorsque n ? +∞, de la suite Sn = n?1∑ j=1 1 j ? log(n) = n?1∑ j=1 (1 j ? log (j + 1 j )) La vitesse de convergence est tres lente, enO(1/n), comme on l'on voit sur le developpement asymptotique ? = Sn + 12n + k∑

• sommes ponderees de s?

• irrationalite

• constante d'euler ?

• xnt dx

• famille classique des polynomes de legendre

• methode particulierement

• ????

• legendre polynomials

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##### Legendre polynomials

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ˆ POLYNOMES DE TYPE LEGENDRE ET APPROXIMATIONS DE LA CONSTANTE D’EULER
T. RIVOAL
R´esume´.propose a simple method to accelerate signiﬁcatively the convergence ofWe Sn=Pnj=111/jlog(n) towards Euler’s constantγ: we construct linear combinations of consecutive values ofSn, to which are assigned certain binomial weights, obtained thanks to a classical integral representation ofγSnand certain special polynomials, of Legendre type. As a special case, we recover an approximation ofγdue to Elsner, obtained by a diﬀerent method. Our approach also applies to the number log(4) which, as Sondow has noted, is in a sense an alternating analogue ofγ; this enables us to produce an apparently new expression ofπ/4 as an inﬁnite product, which can be viewed as an analogue of Vacca’s series forγ. Finally, although this method cannot prove the irrationality ofγ, it is similar to the one used by Alladi and Robinson to prove the irrationality of log(2) by means of Legendre polynomials.
1.Introduction
La constante d’Eulerγmmlelamiti,eolsrestd´eniecoeuqn+, de la suite Sn=jn=X11j1log(n) =jnX=11µ1jlogµj+j1¶¶ Lavitessedeconvergenceesttre`slente,enO(1/ntrled´eveloonpvpoeimtesnumoemnol)c, asymptotique 1k γ=Sn+2n+XB2j2jn12j+R(n, k), j=1 ou`lesB2jsont les nombres de Bernoulli etR(n, k) =O¡(k/πen)2kk/n¢(la constante ` dans leOseitdne´epdnnatedeketn: voir [6]). Aksitnam´xno,eagenepgn´easrmnome´e en revanche le choixn=k2celere`ledluclacc´eaγa`nvcogeeritsuuieqniurfoennetuanss la vitesse 1/k2kmais au prix du calcul des nombres de Bernoulli, qui ne sont pas entiers et dont la croissance est essentiellement factorielle. Sansmˆemeparlerduprobl`emetoujoursouvertdel´eventuelleirrationalite´deγ, on peut plussimplementsedemandercomment,`apartirdelasommeSn, calculer rapidementγ enutilisantdesquantite´scombinatoiresmoinscomplexesquelesnombresdeBernoulli,en particulierdesentiers.Ilexistedenombreusesfac¸onsdere´soudreceprobl`eme(dontcelle deSondow[8]´evoqu´eeplusbas).Uneme´thodeparticulie`rementint´eressanteae´t´edonne´e 1