Stud. Uiv. Babş-Bolyai Math. 560, No., 609 64 O th approximatio of th costat of Napir Adri Vrscu Abstract. Startig from som oldr idas of [] ad [6], w show w facts cocrig th approximatio of th costat of Napir. Mathmatics Subjct Classificatio 00: 6A09, 6D07,40A30, 4A0. Kywords: Th costat of Napir, xpotial fuctio, approximatio, spd of covrgc.. Itroductio Cosidr th two quivalt classical dfiitios of th ral xpotial fuctio x = + x! + x! +... x! +.... rspctivly x = lim + x,. both covrgcs big uiform o compact substs of R. Thir spd of covrgc is diffrt. Cocrig th Taylor-Maclauri approximatio. of th xpotial, s D. S. Mitriović [3], pp. 68-69. For th approximatio giv by., also i this classical book ar giv th followig iqualitis 0 x + x x x, for x ad N ; 0 x x x + x x, for 0 x, N, ; 0 x x x, for 0 x ad N s [4], [5], [3], [4], [5]. I [7] w gav som strogr iqualitis, amly
60 Adri Vrscu i If x > 0, t > 0 ad t > x th x x t + x + max{x, x } x + x t x x t t + x..3 ii If x > 0, t > 0 ad t > x th x x t x + x x x t x x t t x + mi{x, x }.4 ad w dtaild th proof of.3 for th proof of.4 s[], pp. 58-60. Also, ot passat, that th prvious iqualitis giv by th simpl particularizatio x =, th charactrizatios of th spd of covrgc of four stadard squcs rlatd to th umbrs ad, amly + + [8], pag. 38, [] + + + + [0] [6], [7] [6], [7].. Th mai rsult Now w will stablish th bst approximatio of by th family of squcs of gral trm + +p, whr p is a ral paramtr; this may suggst th bst approximatio of x, x > 0, by som algbraic fuctios. Cosidr th kow limitd xpasio + x x = x + 4 x 7 6 x3 + Ox 4,. ad also th limitd biomial o + x p = + p! pp x + x +! pp p x 3 + Ox 4.. 3! Usig th otatios = +, f = + +, g =, h = ad applyig th GM-AM iquality for th umbrs a = a = a 3 =... = a = +, a + =, w obtai that th squc is strictly icrasig s [9]. Applyig th GM-AM iquality for th umbrs b = b = b 3 =... = b =, b + =, w obtai aalogously that th squc g is strictly icrasig. Th idtitis f = ad h g = show us that th squcs f + ad h ar strictly dcrasig. Thrfor f ad g h.
O th approximatio of th costat of Napir 6 Rmark. Th formula. is ca b obtaid i a classical way, usig th wll-kow limitd xpasios l + x = x x + x3 x4 + 3 4 Ox5 ad xp y = + y + y + y3 + y4 + Oy 5. Th!! 3! 4! + x x = xp l + x = x x x = xp l + x = xp + x x3 4 + Ox4 = = 3 k=0 ad som stadard calculatios giv.. k x x + k! x3 4 + Ox4 + Ox 4 Multiplyig. ad., part by part, prformig th usual calculatios ad rplacig x by =,, 3,..., w obtai + +p = + p + p 4p + 4 + + 8p4 36p.3 + 50p 48 3 + O 4. From.3, w s that lim + +p = p 0, for p = for p..4 ad so For p = it rsults that th trm i of.3 vaishs ad w hav + +/ = + 3 + O 4 + which coducts us to th quality lim + = + O, + + =..5 Aothr way to obtai.5 cosists i a rpatd us of th L Hospital s rul, but this givs o ida of th provac of th rsult. So, th bst approximatio of by th squcs of gral trm + +p is th o corrspodig to p =.
6 Adri Vrscu 3. A two-sidd stimat Th quality.5 suggsts us to sarch a two sidd stimat of th form + α + + + β 3. whr α ad β ar two ral costats. Profssor Ioa Gavra commuicatd m [] a covit lft part of 3., amly for α =, w hav + W prst hr his proof. Lt a = + +. 3. + + b ad b = l a, that is b = + [l + l ]. W hav succssivly b = + [ l + + l + ] = + + l + + + l = + l + l + [ = u l + l ], u u whr w hav dotd + = u Usig ow th wll kow xpasios l + x = x x + x3 3... x + + l x = x x x3 3... x
O th approximatio of th costat of Napir 63 uiform covrgt i vry compact K, ad prformig th usual calculatios, w obtai b = + 3 3 + 5 5 +... = + 8 4 +... > bcaus of > 0. Thrfor usig that x > + x, for x > 0 w hav + +/ = a = b > u > + u ad so + +/ > + +, that givs 3.. Th problm of fidig of a adquat costat β i 3. rmais op. 4. Cocludig rmarks Th prvious rsults, cocrig th approximatio of th umbr by th squc + +p coduct to th ida to sarch a similar approximatio of th xpotial. W mtio that a approximatio of th xpotial usig th ratioal fuctios was giv by J. Karamata s []. Ackowldgmts. I thak Profssor Ioa Gavra for his commuicatio of th iquality 3. ad th proof. Rfrcs [] Gavra, I., Privat commuicatio to th author, Cluj-Napoca, Sptmbr 4, 00. [] Karamata, J., Sur l approximatio d x par ds foctios ratiolls, Bull. Soc. Math. Phys. Srbi, 949, o., 7-9. [3] Mitriović, D.S., Aalytic Iqualitis, Sprigr-Vrlag, Brli Hidlbrg Nw York, 970. [4] Nvill, E.H., Not 09: Two iqualitis usd i th thory of th gamma fuctios, Math. Gaztt, 0936, 79-80. [5] Nvill, E.H., Not 5: Additio to th Not 09, Math. Gaztt, 937, 55-56. [6] Niculscu, C.P., Vrscu, A., O th ordr of covrgc of th squc, Romaia, Gaz. Mat., 09004, o. 4. [7] Niculscu, C.P., Vrscu, A., A two-sidd stimat of x + x, JIPAM, 5004, o. 3, art. 55. [8] Pólya, G., Szgö, G., Problms ad Thorms i Aalysis I, Sprigr-Vrlag, Brli Hidlbrg Nw York, 978.
64 Adri Vrscu [9] Siadura, J. St.-C., Applicatios of th iquality of th mas, Math. Gaztt, 4596, o. 354, 330-33. [0] Vrscu, A., A iquality rlatd to th umbr, Romaia, Gaz. Mat., 8798, o. -3, 6-6. [] Vrscu, A., A simpl proof of a iquality rlatd to th umbr, Romaia, Gaz. Mat., 93988, o. 5-6, 06-07. [] Vrscu, A., Th Numbr ad th Mathmatics of th Expotial, Romaia, Publishig Hous of th Uivrsity of Bucharst, 004. [3] Watso, G.N., A iquality associatd with gamma fuctio, Mssgr Math., 4596, 8-30. [4] Watso, G.N., Not 54: Commts o Not 5, Math. Gaztt, 937, 9-95. [5] Whittakr, E.T., Watso, G.N., A Cours of Modr Aalysis, Cambridg Uiv. Prss, 95. Adri Vrscu Valahia Uivrsity of Târgovişt Dpartmt of Scics Bd. Uirii 8 3008 Târgovişt Romaia -mail: avrscu@clickt.ro