Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 1. pp. 13-18, 2011 Applied Mathematics
The Asymptotic Expansion of an Approximation Formula for the Fac-torial Function
Cristinel Mortici
Valahia University of Târgovi¸ste, Department of Mathematics, Bd. Unirii 18, 130082, Târgovi¸ste, Romania
e-mail: cm ortici@ valahia.ro
Received Date: August 31, 2009 Accepted Date: January 27, 2011
Abstract. The aim of this paper is to construct the asymptotic series of the approximation formula ! ≈ r 2 µ + 1 ¶+1 2
stated in Mortici [Arch. Math. (Basel) 93(2009) no. 1 37-45]. Key words: Factorial function; Stirling’s formula; numeric series 2000 Mathematics Subject Classification: 40A25; 26D15; 34E05; 65D20. 1. Introduction
The Stirling’s formula and its different generalizations have a wide class of ap-plications. In consequence, it has been deeply studied due to its practical im-portance. The Stirling’s formula:
(1.1) ! ∼³
e ´√
2
is an approximation for big factorials. In fact, the formula (1.1) was discovered by the French mathematician Abraham de Moivre (1667-1754) in the form
! ≈ constant ·³e´√
while the Scottish mathematician James Stirling (1692-1770) discovered the leading constant√2. For details, [1, 3] can be consulted.
The Stirling’s formula is now the first approximation of the following series (1.2) ! =³ e ´√ 2 exp µ 1 12− 1 3603 + 1 12605 − 1 16807+ ¶ also called the Stirling’s series. For proofs and other details see for example [1]. Recently, Mortici [2] established the following approximation formula:
(1.3) ! ≈ r 2 µ + 1 ¶+1 2
which is stronger than the Stirling’s formula. Associated with the formula (1.3), we define in this paper the asymptotic series
! ∼ r 2 µ + 1 ¶+1 2 exp µ 1 12− 1 122 + 29 3603 − 3 404 + 17 2525− (1.4) − 5 846 + 89 16807 − 7 1448 + ¶
Finally, a systematic way to compute the coefficients of the series (1.4) is pro-vided.
2. The Results
It is introduced in [2] the sequence ()≥1 by the relations
(2.1) ! =³ ´√ 2 exp Ã∞ X = ! ≥ 1
that is, for every ≥ 2
(2.2) = µ +1 2 ¶ ln µ 1 + 1 ¶ − 1
Now let us denote by ()≥1the sequence of the partial sums of the series from
(2.1), = X =1 µµ + 1 2 ¶ ln µ 1 + 1 ¶ − 1 ¶ = X =1
and let be its sum. Then (2.1) becomes
(2.3) ! =³
´√
2 exp ( − −1)
and we are now interested in finding the convergence rate of the sequence ()≥1 More precisely, we have the following high-precision convergence rates:
Theorem 2.1. For the sequence = X =1 µµ +1 2 ¶ ln µ 1 + 1 ¶ − 1 ¶
with the limit we have:
(2.4) lim →∞ (− ) = − 1 12 (2.5) lim →∞ µ (− ) + 1 12 ¶ = 1 12 (2.6) lim →∞ µ µ (− ) + 1 12 ¶ −121 ¶ = −36029 (2.7) lim →∞ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ = 3 40 (2.8) lim →∞ µ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ −403 ¶ = −25217 (2.9) lim →∞ µ µ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ −403 ¶ + 17 252 ¶ = 5 84 lim →∞ µ µ µ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ −403 ¶ + (2.10) +17 252 ¶ −845 ¶ = −168089 lim →∞ µ µ µ µ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ −403 ¶ + (2.11) + 17 252 ¶ −845 ¶ + 89 1680 ¶ = 7 144
From the limit (2.11) it results that there exists a sequence ()≥1
conver-gent to 1 such that µ µ µ µ µ µ µ (− ) + 1 12 ¶ −121 ¶ + 29 360 ¶ −403 ¶ +
+17 252 ¶ − 5 84 ¶ + 89 1680 ¶ =7 144 From here we succesively can deduce that
− = 1 12− 1 122 + 29 3603 − 3 404+ 17 2525− 5 846 + 89 16807 − 7 1448
By replacing − −1= + ( − ) in relation (2.3), we obtain
! =³ ´√ 2· · exp µ + 1 12− 1 122 + 29 3603 − 3 404 + 17 2525 − 5 846 + 89 16807− 7 1448 ¶ or ! =³ ´√ 2 · exp µµ +1 2 ¶ ln µ 1 + 1 ¶ − 1 +121 −1212 + 29 3603− −4034 + 17 2525 − 5 846 + 89 16807 − 7 1448 ¶ or ! = r 2 µ + 1 ¶+1 2 exp µ 1 12− 1 122 + 29 3603 − 3 404 + 17 2525− −8456 + 89 16807 − 7 1448 ¶ The tool for proving the Theorem 2.1 is the following
Lemma 2.1 (Cesaro-Stolz). Let ()≥1 and ()≥1 be two sequences of
real numbers convergent to zero, ()≥1 strictly decreasing, such that exists
the limit = lim →∞ − +1 − +1 Then = lim →∞ Now we are in position to prove the Theorem.
Proof of Theorem 2.1.For the limit (2.4), we have: lim →∞ (− ) = lim→∞ − 1 = lim →∞ (− ) − (−1− ) 1 − 1 −1 = = lim →∞ ¡ +1 2 ¢ ln¡1 + 1 ¢ − 1 1 − 1 −1 = −121
For the limit (2.5), we have: lim →∞ µ (− ) + 1 12 ¶ = lim →∞ (− ) +121 1 = lim →∞ (− ) + 121 1 2 = = lim →∞ ¡ (− ) +121 ¢− ³ (−1− ) +12(1−1) ´ 1 2 −(−1)1 2 = = lim →∞ (− −1) +121 − 1 12(−1) 1 2 −(−1)1 2 = 1 12
Even if these limits can be obtained directly where some difficulties appear, we had the idea to use Maple for a simple calculation of this limits and for the next more complicated limits. The next limits (2.6)-(2.11) are practical imposible to be computed without Maple or other similar program.
As we have already seen, before using the Lemma 3.1, we always simplify the fraction to bring our limit in the following advantageous form:
lim →∞ (− ) − 1
( is a notation for the remaining expression) then we apply the Lemma 3.1:
lim →∞ ((− ) − ) − ((−1− ) − −1) 1 −(−1)1 = = lim →∞ (− −1) − + −1 1 −(−1)1
The last limit is a limit involving rational functions and a logarithm function which is easy to compute using Maple.
The other limits (2.6)-(2.11) are computed in a similar way and we will omit here these computations for sake of simplicity.¤
Now we are concerned to give a systematical way to find the coefficients of the series (1.4). We start from [1, Rel. 6.1.40]:
ln Γ () ∼ µ −12 ¶ ln − + ln√2 + ∞ X =1 2 2 (2 − 1) 2−1
where denote Bernoulli numbers, see [1, p.807]. We set = + 1 and obtain
ln ! = µ +1 2 ¶ ln ( + 1) − − 1 + ln√2 + where (2.12) ∼ ∞ X =1 2 2 (2 − 1) ( + 1)2−1
This is an asymptotic expansion in powers of +11 In order to obtain an expan-sion in powers of 1 we use
1 ( + 1) ∼ ∞ X = (−1)+ µ − 1 − ¶ 1
Then (2.12) yields after a simple rearrangement
(2.13) ∼ ∞ X =1 (−1)−1 [+1 2 ] X =1 µ − 1 2 − 2 ¶ 2 2 (2 − 1) Using [1, Rel. 23.1.7], we can simplify (2.13) to
∼ ∞ X =1 where = ∙ (−1)+1+1+ − 1 2 ¸ (−1)+1 ( + 1) In conclusion, ! ∼ r 2 µ + 1 ¶+1 2 exp (∞ X =1 ∙ (−1)+1+1+ − 1 2 ¸ (−1)+1 ( + 1) ) References
1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1965.
2. C. Mortici, An ultimate extremely accurate formula for approximation of the fac-torial function, Arch. Math. (Basel), 93(2009), no. 1, 37-45.
3. J. O’Connor, E. F. Robertson, James Stirling, MacTutor History of Mathematics Archive.