Approximating the Powers with Large Exponents and Bases Close to Unit, and the Associated Sequence of Nested Limits

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1 In. J. Conemp. Ma. Sciences Vol no Approximaing e Powers wi Large Exponens and Bases Close o Uni and e Associaed Sequence of Nesed Limis Vio Lampre Universiy of Ljubljana Slovenia 386 vio.lampre@fgg.uni-lj.si Absrac Te rae of convergence of lim 1+ x is esimaed accuraely and e associaed sequence of nesed limis saring wi e limi is esablised. [ lim 1+ x e x] = 1 2 ex x 2 Maemaics Subjec Classificaion: 26D7 4A25 Keywords: approximaion convergence exponenial funcion nesed limis sequence 1 Inroducion In e maemaical lieraure we frequenly mee e expressions like 1 + x wic sould be esimaed for given x R and large see [1] for example. Moreover in [3] e Wallis produc W n := n 4k 2 4k 2 1 was presened in e form W n = 1 2 πe 1 1 Θn 2n+1 exp 2n +1 6n Θ n 6n +3 k=1

2 2136 V. Lampre and consequenly e expression π = W n e Θ 2n+1 exp n 2n 6n +3 Θ n 6n is given for every posiive ineger n wi Θ n Θ n 1. So we are ineresed primarily ow o esimae e power 1+ x for given x R and large R +. Te rae of monoonous convergence of lim 1+ x was esimaed in [2] as exp x x 2 < 1+ x < exp x 21 ε x ε valid for every real x ε 1 and x /ε. In e same paper e rae of convergence of [ lim e x 1+ x ] = ex x was also examined. Te formula 1 represens an approximaions for e funcion 1+ x useful for very large. In wa follows we sall esablis more accurae approximaions using only elemenary ecniques and we sall also answer e quesion { [ e x x 2 lim e x 1+ x ]} =? 2 posed in [2]. In addiion we sall find furer nesed limis saring wi 2. 2 Preliminaries For given x R {} suc a > x we ave 1+ x = exp ϕx =: Gx 3 were ϕ x 1 ln1 + x 4 wi = 1 x being a parameer. If < x < 1 e funcion ϕ x is well defined on e puncured inerval 1/ x 1/ x and we ave for any ineger

3 Approximaing e powers wi large exponens 2137 m ϕ x = 1 dτ 1+τ = 1 m j= τ j + τn+1 1+τ dτ = S m x +R m x 5 were S m x = m 1 j x j+1 j 6 j +1 j= R m x = 1m+1 τ n+1 dτ 1+τ. 7 For e remainder R m x we ave if < x < 1 R m x 1 x s m+1 ds 1 x = x m+2 m + 21 x as m. 8 Hence ϕ x = 1 j x j+1 j 9 j +1 j= for < < 1/ x. Terefore ϕ x is represened by a funcion wic is analyic on e disk < 1/ x. Addiionally we define ϕ x = x e sum of e series 9 and Gx = e x. Consequenly ϕ x C 1/ x 1/ x and referring o 3 e same is rue for e funcion exp ϕ x. Hence for G considered as e funcion of e variable we ave G C 1/ x 1/ x Gx = 1 + x 1/ for and Gx = e x. 1 According o 9 e derivaives are given as ϕ j x = 1j j! j +1 xj+1 11

4 2138 V. Lampre for j 1. Consequenly we obain G x = 1 2 ex x 2 2 G 2 x = 1 12 ex 3x 4 +8x 3 3 G x = ex x 6 +8x 5 +12x G 1 x = 4 24 ex 15x x x x 5 5 G 5 x = 1 96 ex 3x 1 +8x x x x 6 for example. In paricular cases considering e sign of e produc x e remainder R m x in 7 can be esimaed more precisely. We disinguis wo possibiliies: x > and x <. A <x<1. In is case we esimae τ m+1 dτ 1+τ τ m+1 dτ 1+τ > < τ m+1 dτ 1+x = x m+2 m x τ m+1 dτ = xm+2 m +2. Consequenly using 7 we obain x n+2 m x < 1m+1 sgn R m x < xm+2 m B 1 <x<. Now we ave referring o 7 R m x = 1m+1 x s m+1 ds 1 s = 1 x s m+1 ds 1 s. Terefore proceeding as above we find x m+2 m +2 < sgn R mx < x m+2 m +2 1 x Approximaing e funcion 1+ x How o esimae e expression 1+ x answers e following eorem.

5 Approximaing e powers wi large exponens 2139 Teorem 1. For any ineger m for x R {} and for > x we ave e equaliy 1+ x = ex exp s m x +r m x 15 were 1 s m x = m j=1 1 j xj+1 j +1 j = m+1 k=2 1 k xk k k+1 16 and e remainder r m x is esimaed in e following way: 1 2 x m+2 m x m < 1m+1 r m x < x m+2 if x> 17 m +2m+1 x m+2 m +2 x < r mx < x m+2 for x<. 18 m m +2m+1 Proof. Le m x and fulfil all e condiions of e eorem. Ten considering 3 and subsiuing = 1 in 5 we obain e expression 15 saisfying 16. Ten using 13 we find e esimae 17 and referring o 14 we ge e inequaliies 18. Seing m = 1 2 in e preceding eorem we obain e following wo corollaries. Corollary 1.1. For x> and >xe following inequaliies old e x exp < 1+ x <e x exp x2 e x exp x2 + 2 e x exp x2 + x x2 2 x3 3+x x4 4 3 < 2+x < 1+ x <e x exp x2 + x x <ex exp x2 2 + x3 3 2 Corollary 1.2. For x< and > x we ave e inequaliies e x exp x2 < 1+ x <ex exp e x exp x2 2 2 x x 3 3 x e x exp x2 x 3 x x 1 By definiion q j=p = for p>q. x2 2 < 1+ x <ex exp x2 x 2 2 x x 3 2 < 1+ x <ex exp x2 2 x 3 x

6 214 V. Lampre Consequenly insering x = in e corollaries above we ge e following corollary. Corollary 1.3. For > and < 1 we ave e esimaes e exp 2 < 1 + <e exp e exp 2 < 1 + <e exp for <<1 for 1 <<. Example. Using e Corollaries 1.1 and 1.2 we obain for >1 e following esimaes: e exp 1 < 1+ 1 <e exp e exp 1 < 1 1 < e exp 1 2 e exp < < e exp exp < e 8 +4 From e second inequaliy i also follows 2 < e exp 1 e exp < 1 1 <e exp Corollary 1.4. For >x> ere olds e esimae e x > e x := e x < e x := x [1+ x2 2 x3 3 + x x [1+2 x2 x 2 for > x 2 2 x x4 4 3 ] ]. Proof. Using Teorem 1 wi m = 2 we ave e x = 1+ x exp s2 x r 2 x 19 were s 2 x = x2 2 x3 3 2

7 Approximaing e powers wi large exponens 2141 and Tus for >x> x x 2 < r 2x < x s 2 x r 2 x > x2 2 x x x 2 > and s 2 x r 2 x < x2 2 x3 3 + x = x x 2 x x + x x < x x 2 x x + x x 3 4 < x x 2. Tese esimaes ogeer wi e expression 19 and e inequaliies 1 + < e < 1+e <1+4 valid for > verify e corollary. Figure 1 illusraes e raional approximaions o e x given in Corollary 1.4 for x = e e 1n e 1n n Figure 1: Te graps of e sequences n e 1n and n e 1n; Corollary Nesed limis For e funcion 1+ x =: Ex 2

8 2142 V. Lampre x R being a parameer we define E 1 x E 2 x ec... by E 1 x := [ Ex e x] e 1 x = lim E 1 x 2 = ex x 2 2 E 2 x := [ E 1 x e 1 x ] e 2 x := lim E 2 x ec... Here we ave wo quesions abou e sequence e n x : is exisence and n N is compuaion. In order o solve is problem we sall use e following lemma. Lemma 1. Le r R + e inerval I = r r f C I and for any n N f n+1 = Ten e sequence f n n N n N we ave f n = f = f 1 = f f <r f n+1 n + 1! = f n f n <r is well defined a any I. Moreover for every n f n 1 wi f n differeniable a =and f n = n! f i i <r i= i! 23 f n+1 = f n = f n+1 n + 1!. 24 Proof. We can verify e equaliies 23 by inducion. Ten considering 23 and using e Taylor formula of order n we ave for any n N and I {} f n+1 = f n f n = f n+1 ϑ n + 1! for some ϑ = ϑn 1. Since f C I we ge f n+1 = f n = lim f n f n = f n+1 n + 1!.

9 Approximaing e powers wi large exponens 2143 Now we are in a posiion o formulae e following eorem abou e nesed limis. Teorem 2. For any n N x R \{} and > x le Ex := 1+ x and E 1 x := [ Ex e x] e n x := lim E nx E n+1 x := [ E n x e n x ]. Ten e sequences E n x and e n N n x are well defined for any > n N x and we ave see 3 and 12 e n x = 1 n! n G x for n N and x R \{}. n Proof. We fix x R {} deermining e inerval I = r r wi r =1/ x. Nex we consider e funcion f f Gx. According o 1 we ave f C I. Referring o Lemma 1 f generaes e sequence f n suc n N a e relaions and 24 old. Referring o 1 and 21 we obain for > x E 1 x = Ex e x = Gx 1/ Gx = f1/ f 1/ = f 1 1/ and defining f :=f e 1 x = lim E 1 x =f 1 = f = f. Generally for n N and > x we ave E n x =f n 1/ and e n x =f n = f n Indeed if 25 is rue en E n+1 x = [ E n x e n x ] = f n1/ f n 1/ = f n+1 1/. Consequenly considering e coninuiy differeniabiliy of f n e n+1 x = lim E n+1x = lim f n+11/ =f n+1 = f n. Hence e sequences E n x and e n N n x n N > x > and referring o 25 and 24 we ge are well defined for any e n x = f n n! = 1 n! n G x. n

10 2144 V. Lampre From Teorem 2 and relaions 12 e following Corollary follows. Corollary 2.1. For x we ave e 1 x = 1 2 ex x 2 e 2 x = 1 24 ex 3x 4 +8x 3 e 3 x = 1 48 ex x 6 +8x 5 +12x 4 e 4 x = ex 15x x x x 5 e 5 x = ex 3x 1 +8x x x x 6. Figure 2 sows e graps of e funcions x e n x from Corollary 2.1. n 3 n 4 n e n x n 2 n 4 x n 2.1 n Figure 2: Te graps of e funcions x e n x. Open quesion. Find e closed-form formula for e sequence n e n x i.e. find e closed-form formula for e sequence n P n x of polynomials suc a e n x e x P n x. References [1] H. J. Broers and J. A. Knox New Closed-Form Approximaions o e Logarimic Consan e Ma. Inelligencer pp [2] V. Lampre Esimaing Powers wi Base Close o Uniy and Large Exponens Divulgaciones Maemáicas pp [3] V. Lampre Wallis sequence esimaed roug e Euler-Maclaurin formula: even from e Wallis produc π could be compued fairly accuraely Ausral. Ma. Soc. Gaz pp

11 Approximaing e powers wi large exponens 2145 [4] S. Wolfram Maemaica version 7. Wolfram Researc Inc Received: May 211