Some intriguing limits : continuation; O pewnych intrygujących granicach : kontynuacja - Digital Library of the Silesian University of Technology

Seria: MATEMATYKA STOSOWANA z. 3 Nr kol. 1899

Piotr LORENC, Roman WITUŁA Institute of Mathematics

Silesian University of Technology

SOME INTRIGUING LIMITS – CONTINUATION

Summary. Wituła and Słota in [College Math. J. 42 (2011), 328] pro- posed a way of proving the relation (1) given below which appeared to be a genuine result. Authors of the present paper, inspired by the form of this limit, try to find some generalizations of this one, also in the context of some special functions (e.g. the gamma function, the generalized Laguerre polynomials).

O PEWNYCH INTRYGUJĄCYCH GRANICACH – KONTYNUACJA

Streszczenie. Wituła i Słota w notce [College Math. J. 42 (2011), 328]

zaproponowali udowodnienie relacji (1), podanej poniżej, która wydaje się bardzo ciekawą zależnością. Autorzy niniejszego artykułu, zainspirowani po- stacią tej granicy, próbują znaleźć różne jej uogólnienia także w kontekście pewnych funkcji specjalnych (np. funkcji gamma, uogólnionych wielomia- nów Laguerre’a).

2010 Mathematics Subject Classification: 40A05.

Corresponding author: R. Wituła (Roman.Witula@polsl.pl).

Received: 06.11.2013 r.

Piotr Lorenc will be next year the MSc graduate student in Mathematics.

Problem of evaluating the limits of functions depending on the expressions (1 +^α_x)^x(x → ∞) is like never ending story. Look at the following ones (see [15]):

x→∞lim

 e⁽⁻¹⁾

n n

 . . .

e¹⁴ e⁻¹³

e¹² e⁻¹

1 + 1 x

xxxxx

. . .x

| {z }

(n+1)−times

= e^(−1)n+1n (1)

for every n ∈ N;

x→∞lim

exp(An) . . .

exp(A2)

exp(A1) Qr i=1

1 +^α_xⁱ
βix

Qr j=1

1 + ^γ_x^j
δjx

xx

. . .

xx

| {z }

n−times

=

= exp(−Aⁿ⁺¹), (2) for every n, r ∈ N, αi, β_i, γ_i, δ_i∈ R, i = 1, . . . , r where

A_k := 1 k

αk δ γk β

:= 1

k(αk◦ β − γk◦ δ) , k∈ N,

and the symbol ◦ denotes the scalar product applied to the vectors αk,β, γ_k,δ∈ R^r, defined in the following way

αk := [(−α¹)^k,(−α²)^k, . . . ,(−α^r)^k], β:= [β1, β2, . . . , βr], γk := [(−γ¹)^k,(−γ²)^k, . . . ,(−γ^r)^k], δ:= [δ1, δ2, . . . , δr];

x→∞lim

2 α β x (α − β) e

1 + 1 α x

αx

− 1 + 1

β x

βx!^x

= exp



−11 12

1 α+1

β

 (3)

for every α, β ∈ R \ {0}, α 6= β;

x→∞lim ln

 ln

1 + 1 x

x_−2x!x

= exp

− 5 12

; (4)

x→∞lim x

 e−

1 + 1 x

x

= e

2; (5)

x→∞lim x²

 e− e

2x −

1 + 1 x

x

= −11

24e. (6)

Sketch of the proofs of (1) and (3).

We have (x > 1):

1 + 1 x

x

= exp

 xln

1 + 1 x

= exp X^∞

n=0

(−1)ⁿ (n + 1) xⁿ

!

which implies (1). Now, let α, β ∈ R \ {0}, α 6= β. Then we get

1 + 1 αx

αx

− 1 + 1

βx

βx

= exp α xln

1 + 1 αx

− exp β xln

1 + 1 βx

=

= exp 1 − 1

2αx+ 1

3α²x²+ o1 x²

− exp 1 − 1

2βx + 1

3β²x² + o 1 x²

=

= exp 1 − 1

2βx + 1

3β²x² + o 1 x²



×

×h

exp 1 2βx −

1

2αx+ 1

3α²x² − 1

3β²x² + o 1 x²



− 1i

=

= e exp

− 1

2βx+ o1 x

h α − β

2αβx −(α − β)(5α + 11β) 24α²β²x² + o 1

x²

i ,

which implies

 2αβx (α − β)e

h1 + 1 αx

αx

− 1 + 1

βx

βxi^x

=

= exp

− 1

2β+ o(1)

1 − 5α + 11β 12αβx + o1

x

x

,

which easily proves (3). 

Proof of (2).

We have (x > max{|α|, |β|}):

1 + ^α_x
βx

1 +^γ_x
δx = exp

 β xln

1 + α x

− δ x ln 1 + γ

x

=

= exp

 ∞

X

n=0

(−1)ⁿ β αⁿ⁺¹− δ γⁿ⁺¹
1 (n + 1)xⁿ

 ,

which implies (2). 

Sketch of the proofs of (5) and (6).

By substitution _x¹ = t we get

x→∞lim x

 e−

1 + 1 x

x

= lim

t→0⁺

e− (1 + t)^1t

t ,

x→∞lim x²

 e− e

2x −

1 + 1 x

x

= lim

t→0⁺

e−^e2· t − (1 + t)^1t

t² .

Moreover we have (x > 1 ⇒ t ∈ (0, 1)):

(1 + t)^1t = exp1

t ln(1 + t)

= expX^∞

n=0

(−1)ⁿ n+ 1tⁿ

. (7)

Application of l’Hospital’s rule and relation (7) gives the values of both limits.  Proof of (4).

We have

ln 1 + 1

x

x

= 1 − 1 2x+ 1

3x² + o 1 x²

 and

ln

 1 − 1

2x + 1

3x² + o 1 x²

^−2x

= 1 − 5

12x+ o1 x

,

which implies (4), since

x→∞lim

1 − 5

12x+ o1 x

x

= exp

− 5 12

 .

 Remark 1.We note that limit (1) for n = 1 is ”a regular guest” of many problems in calculus books (see e.g [4, 6, 8]). In contrast, the limit of general shape (1) and its generalizations (2) are probably new.

Remark 2.Limits (3)–(6), which are probably new (see e.g. [4, 6–8, 10, 11, 15]), arose during the discussion on generalizations of limits (1) and (2) (and they are far from our expectations). Authors hope that these limits inspire the Readers to look for another results.

Remark 3.We note that

x→+∞lim x

1 + 1 x− α

x

− 1 + 1

x

x

= α e for every α ∈ R.

Proof. From the following formula

1 +a x

x

= exp

 xln

1 + a x

= exp(a) exp X^∞

n=1

(−1)ⁿaⁿ⁺¹ (n + 1) xⁿ

!

=

= e^a X^∞

k=0

1 k!

X∞ n=1

(−1)ⁿaⁿ⁺¹ (n + 1) xⁿ

k!

=

= e^a

 1 − a²

2 x+ a 8 +1

3

 a³ x² − a²

24+a 3 +1

2

 a⁴ 2 x³+ . . .

 , for a = 1 and from the following decomposition (which is a special form of binomial series):

1 + 1 x− α

α

= 1 + α

x− α+ o 1 x− α

 we get

x

1 + 1 x− α

x

− 1 + 1

x

x

=

= 1 + 1

x− α

α

x

1 + 1 x− α

x−α

− e

 + x

 e−

1 + 1 x

x + + e x

1 + 1 x− α

α

− 1



x→∞−→ −e 2 +e

2+ α e = α e.



Remark 4.In paper [5] the authors observed that from formula (1) for n = 1 and from Stirling’s formula (see e.g. [9]) the following approximation holds

Γ(x + 1) ∼r 2πx

e · x^x2+x (x + 1)^x2.

Starting from this formula Feng and Wang (authors of [5]) proved that for suffi- ciently large x ∈ R the following one holds

Γ(x + 1) =√

2π · x^x+12 x − 1 x+ 1

^x22+

Pm k=0

αk x2k+O



1 x2m+2

 ,

where constants α_k satisfy the recurrence relation Xk

j=0

αj

2k − 2j + 1 = − 1 2(2k + 3) −

B2k+2

2(2k + 1)(2k + 2)

for k = 0, 1, 2, . . . and Bn denotes the n-th Bernoulli number (see e.g. [9]).

Remark 5.In paper [13] the asymptotic expansion of the Gamma functions ratio is obtained

Γ(z + α)

Γ(z + β) ∼z^α−β 1 + (α − β)(α + β − 1) 1 2z+ +α − β

2

3(α + β − 1)²− α + β − 1 1

12z²+ O z⁻³

! ,

where α, β, z ∈ C and z → ∞. From this expansion we deduce the relation

z→∞lim z^β−αΓ(z + α) Γ(z + β)

!z

=1

2(α − β)(α + β − 1), where each power has its principal value.

Remark 6.In paper [14] the formulae of type (1) connected with limit

n→∞lim

√n

n!

n = e⁻¹

are discussed. For example, there are presented the following relations

n→∞lim e⁻¹²¹



e· ^n√_n^n!n

√2πn

!n

= 1

and

n→∞lim

√n!

2π

e q

n²+ n +¹₆

!n+¹₂!n³

= e²⁴⁰¹ .

Remark 7.Let L^(−a)n (−z) denote the n-th generalized Laguerre polynomial (see [12]). Then we obtain the relation (see [1–3]):

n→∞lim L^(−a)_n (−z) · e^−z2 2√π ·

e^2√nz z^14−a2n^14+a2

₋₁!^√n

=

= 1

48√z

3 − 12a²+ 24(1 − a)z + 4z²

for z ∈ C \ (−∞, 0], a ∈ C, where Re(√nz) denotes p

n|z| cos

θ 2

 and θ :=

arg(z) ∈ (−π, π].

References

1. Assche van W.: Weighted zero distribution for polynomials orthogonal on an infinite interval. SIAM J. Math. Anal. 16 (1985), 1317–1334.

2. Assche van W.: Erratum to ”Weighted zero distribution for polynomials or- thogonal on an infinite interval”. SIAM J. Math. Anal. 32 (2001), 1169–1170.

3. Borwein D., Borwein J.M., Crandall R.E.: Effective Laguerre asymptotics.

SIAM J. Numer. Anal. 46 (2008), 3285–3312.

4. Dorogovcev A.J.: Mathematical Analysis – Problems. Wisza Szkola, Kiev 1987 (in Russian).

5. Feng L., Wang W.: Two families of approximations for the gamma function.

Numer. Algorithms 64 (2013), 403–416.

6. Kudravcev L.D., Kutasov A.D., Czechlov W.I., Szabunin M.I.: Problems in Mathematical Analysis (Limits, Continuity, Differentiability). Phys.-Mathem.

Literature Press, Moscov 1984 (in Russian).

7. Lorenc P., Wituła R.: Limits and functions connected with e (in preparation).

8. Polya G., Szeg¨o G.: Aufgaben und Lehrs¨atze aus der Analysis. Erster Band, Springer, Berlin 1964.

9. Rabsztyn Sz., Słota D., Wituła R.: Gamma and Beta Functions, Part I. Wyd.

Pol. Śl., Gliwice 2012 (in Polish).

10. Sadovniczij W.A., Grigorian A.A., Koniagin S.W.: Problems for Students Ma- thematical Olympiads. Moscov University Press, Moscov 1987 (in Russian).

11. Sadovniczij W.A., Podkolzin A.S.: Problems for Students Mathematical Olym- piads. Nauka Press, Moscov 1978 (in Russian).

12. Szeg¨o G.: Orthogonal Polynomials. American Mathematical Society, Provi- dence 1975.

13. Tricomi F., Erdelyi A.: The asymptotic expansions of a ratio of gamma func- tions. Pacific J. Math. 1 (1951), 133–142.

14. Wituła R., Jama D., Nowak I., Olczyk P.: Variations on sequences of arith- metic and geometric means. Zeszyty Nauk. Pol. Śl. Mat. Stosow. 1 (2011), 81–98.

15. Wituła R., Słota D.: Intriguing limit. College Math. J. 42 (2011), 328.

Omówienie

Autorzy prezentowanego artykułu, zainspirowani granicą (1) zamieszczoną w notce [College Math. J. 42 (2011), 328], przedstawiają tu wiele różnych no- wych granic. Niektóre z nich to uogólnienia wspomnianej granicy (1), a inne sta- nowią jedynie próbę ujęcia takich uogólnień. Obszar badań i dyskusji rozszerzono na funkcje specjalne, między innymi funkcję gamma, oraz uogólnione wielomiany Laguerre’a.