A Refinement of Ramanujan's Factorial Approximation

A refinement of Ramanujan’s factorial approximation Michael D. Hirschhorn1 and Mark B. Villarino2 1 School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia 2 1 Escuela de Matemática, Universidad de Costa Rica San José 11501, Costa Rica Introduction All we have of Ramanujan’s work in the last year of his life is about 100 pages (probably a small fraction of his final year’s output), held by Trinity College, Cambridge, and named by George E. Andrews “Ramanujan’s Lost Notebook”. It was published in photocopied form [4]. In it, Ramanujan [4, p. 339] makes the claim that Γ(x + 1) = √  x  1 x θx 6 3 2 π 8x + 4x + x + e 30 3 < θx < 1, and gives some numerical evidence for where θx → 1 as x → ∞ and 10 this last statement. Inspired by this, we confine ourselves to the positive integers, and prove the following stronger result. Theorem 1. Let the function, θn , be defined for n = 1, 2, . . . by the equation: n! := √  n  1 n θn 6 3 2 π 8n + 4n + n + . e 30 Then, the correction term θn (a) satisfies the inequalities: 1− 79 11 79 20 11 + < θn < 1 − + + ; 2 2 8n 112n 8n 112n 33n3 (1) (b) is an increasing function of n; and (c) is concave, that is, θn+1 − θn < θn − θn−1 . The inequalities (1) are new. In 2006, Hirschhorn [2] proved a less exact version of the inequalities (1). In 2001, Karatsuba [3] proved Ramanujan’s approximation and gave a proof, quite different from ours, of the monotonicity of the correction 1 term θx , for all real x ≥ 1, a result which is stronger than ours. Moreover, although Karatsuba derived an asymptotic expansion for θx , including a uniform error term, she did not derive any explicit numerical inequalities, as we do. Then, in 2003, Alzer 1 [1] proved that in (0, 1] the constant term 100 can be replaced by the best possible min0.6≤x≤0.7 θx = 0.0100450 · · · and that the improved double inequality for θx holds for 0 ≤ x < ∞. The monotonicity of θn was proved by Villarino, Campos-Salas, and Carvajal-Rojas in [5] as a simple consequence of the inequality in [2]; in that paper, the concavity of θn was also noted, without proof. Our proofs use nothing more than the series for log (1 + x) and exp{x}. We will find it convenient to use the following notation. Definition 2. The notation Pk (n) means a polynomial of degree k in n with all of its non-zero coefficients positive. 2 The proofs Proposition 3. The following inequality is valid for n = 1, 2, . . . :     1 1 1 1 log 1 + −1< − . 0< n+ 2 n 12n 12(n + 1) (2) Proof. We have, for |x| < 1, log (1 + x) = x − x2 x3 + − ··· . 2 3 It follows that for |x| < 1,     x3 x5 1+x =2 x+ + + ··· . log 1−x 3 5 1 where n ∈ Z+ , we obtain 2n + 1     1 1 1 1 =2 + + + ··· . log 1 + n 2n + 1 3(2n + 1)3 5(2n + 1)5 If we set x = (3) It follows that     1 1 1 1 n+ log 1 + =1+ + + ··· 2 2 n 3(2n + 1) 5(2n + 1)4 Therefore     1 1 1 0 < n+ · log 1+ −1 < 2 n 3(2n + 1)2 1 1 1− (2n + 1)2 = 1 1 − . 12n 12(n + 1) The inequality (2) leads to the following well-known version of Stirling’s formula. 2 Proposition 4. The following inequality is valid for n = 1, 2, . . . :  n  n   √ √ n n 1 2πn . < n! ≤ 2πn exp e e 12n Proof. Let (4) . √  n n . n an = n! e Then an = an+1  1 1+ n n+ 1 2 e = exp  1 n+ 2   1 log 1 + n   −1 . From (2) we have   an 1 1 1< < exp − . an+1 12n 12(n + 1) (5) So an is decreasing and, if we write 1, 2,. . . , n − 1 for n and multiply the results, we find     1 1 1 a1 < exp − < exp , an 12 12n 12 or,     1 11 = exp . an > a1 exp − 12 12 It follows that a∞ = limn→∞ an exists, and   11 . a∞ ≥ exp 12 √ In fact, from Wallis’s product, a∞ = 2π. If in (5) we write n, n + 1,. . . , N − 1 for n, multiply the results, let N → ∞ and use Wallis’s product, we obtain, successively,   1 1 an < exp − , 1< aN 12n 12N   1 1 aN < an < aN exp − , 12n 12N   √ √ 1 2π < an ≤ 2π exp 12n and √    n  n √ 1 n n . < n! ≤ 2πn exp 2πn e e 12n We shall improve on (2), and so also on (4), by proving the next inequality. 3 Proposition 5. The following inequality is valid for n = 1, 2, . . . :       1 1 1 1 1 1 1 1 1 + − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5   1 1 1 − − 1680 n7 (n + 1)7     1 1 < n+ log 1 + −1 2 n       1 1 1 1 1 1 1 1 1 + . − − − − < 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5 Proof. We have, from (3),     1 1 1 1 1 n+ + + · log 1 + −1< 2 4 2 n 3(2n + 1) 5(2n + 1) 7(2n + 1)6 (6) 1 1− 1 (2n + 1)2 1 1 1 + + 2 4 3(2n + 1) 5(2n + 1) 28n(n + 1)(2n + 1)4       1 1 1 1 1 1 1 1 1 + − − − − = 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5 = 163n6 + 489n5 + 604n4 + 393n3 + 141n2 + 26n + 2 2520n5 (n + 1)5 (2n + 1)4       1 1 1 1 1 1 1 1 1 < + − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5 − and     1 1 1 1 1 1 n+ log 1 + −1> + + + 2 4 6 2 n 3(2n + 1) 5(2n + 1) 7(2n + 1) 9(2n + 1)8       1 1 1 1 1 1 1 1 1 = + − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5   1 1 1 − − 7 1680 n (n + 1)7 P12 (n) + 5040n7 (n + 1)7 (2n + 1)8       1 1 1 1 1 1 1 1 1 > + − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5   1 1 1 . − − 1680 n7 (n + 1)7 This completes the proof. We now demonstrate the greatly improved version of Stirling’s formula. Proposition 6. For n = 1, 2, . . . , the following inequality is valid:  n   √ n 1 1 1 1 2πn − + − exp e 12n 360n3 1260n5 1680n7    n √ 1 1 1 n . − + exp ≤ n! ≤ 2πn e 12n 360n3 1260n5 4 (7) Proof. It follows from (6) that        1 1 1 1 1 1 1 1 1 exp + − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5   1 1 1 − − 1680 n7 (n + 1)7 an < an+1        1 1 1 1 1 1 1 1 1 < exp + . − − − − 12 n n + 1 360 n3 (n + 1)3 1260 n5 (n + 1)5 Thus, for N > n,          1 1 1 1 1 1 1 1 1 1 1 1 exp + − − − − − − 12 n N 360 n3 N 3 1260 n5 N 5 1680 n7 N 7        an 1 1 1 1 1 1 1 1 1 < + , < exp − − − − aN 12 n N 360 n3 N 3 1260 n5 N 5 and √  n   n 1 1 1 1 2πn + − − exp e 12n 360n3 1260n5 1680n7    n √ 1 1 1 n . − + exp ≤ n! ≤ 2πn e 12n 360n3 1260n5 1 Extracting the fraction from the exponents, we see that we can write this last 6 inequality in the form √  n   1  6 n 1 1 1 1 2πn − + − exp 3 5 7 e 2n 60n 210n 280n  1   n  √ 6 1 1 1 n exp − + . ≤ n! ≤ 2πn e 2n 60n3 210n5 We now obtain upper and lower bounds for these new exponents. Proposition 7. The following inequalities are valid for n ≥ 2:   1 1 1 1 − + − exp 2n 60n3 210n5 280n7 1 1 1 11 79 >1+ + 2+ − + 3 4 2n 8n 240n 1920n 26880n5 (8) and   1 1 1 exp − + 2n 60n3 210n5 1 1 1 11 79 1 <1+ + 2+ − + + . 3 4 5 2n 8n 240n 1920n 26880n 396n6 (9) Assuming for the moment that these bounds are valid, we can now prove the main result of this paper. 5 Proof of Theorem 1. It follows from Proposition 7 that for n ≥ 2, 1  n  √ 6 1 1 1 11 79 n 1+ + 2+ − + 2πn 3 4 5 e 2n 8n 240n 1920n 26880n 1  n  √ 6 1 1 1 11 79 1 n 1+ + 2+ − + + , < n! < 2πn e 2n 8n 240n3 1920n4 26880n5 396n6 or:     1  6 √ n n 79 1 11 3 2 π + 8n + 4n + n + 1− 2 e 30 8n 112n  n   1  6 √ n 79 20 1 11 3 2 + + 8n + 4n + n + 1− . < n! < π e 30 8n 112n2 33n3 This beautiful formula is the refined estimate (1). It is easy to check this for n = 1 also, so we have the desired result. To show that θn is increasing, from (1) it follows that   11 79 11 79 20 θn+1 − θn > 1 − + − 1− + + 8(n + 1) 112(n + 1)2 8n 112n2 33n3 = (5082n2 + 7792n + 8497)(n − 2) + 14754 3696n3 (n + 1)2 >0 for n ≥ 2, and it is easily checked for n = 1 also, so θn is increasing. Finally, to prove the concavity of θn , we note that: θn+1 − 2θn + θn−1 79 20 11 + + <1− 8(n + 1) 112(n + 1)2 33(n + 1)3   11 79 20 11 79 +1− + + −2 1− + 8(n − 1) 112(n − 1)2 33(n − 1)3 8n 112n2 (2842n4 + 6389n3 + 15061n2 + 85733n + 433747)(n − 5) + 2166128 1848n2 (n − 1)3 (n + 1)3 < 0 for n ≥ 5, =− and is easily checked for n = 2, 3 and 4 also. We complete the proof of the exponential inequalities as follows. Proof of Proposition 7. Let q := 1 1 1 1 − + − . Then q > 0, and 3 5 2n 60n 210n 280n7 q2 q3 q4 q5 q + + + + 1! 2! 3! 4! 5! 1 1 1 11 79 =1+ + 2+ − + 3 4 2n 8n 240n 1920n 26880n5 P28 (n)(n − 2) + 5421638789368547485949 + 50185433088000000n35 1 1 11 79 1 + 2+ − + >1+ 3 4 2n 8n 240n 1920n 26880n5 exp{q} > 1 + 6 which proves (8) for n ≥ 2. Now let r := 1 1 1 − + . Then r > 0, and 3 2n 60n 210n5 r r2 r3 r4 r5 r6 r7 + + + + + + + ··· 1! 2! 3! 4! 5! 6! 6!  r2 r3 r4 r5 r6 r + + + + =1+ + (1 − r) 1! 2! 3! 4! 5! 6! 1 1 11 79 1 1 =1+ + 2+ − + + 3 4 5 2n 8n 240n 1920n 26880n 396n6 P23 (n)(n − 3) + 239259521624400145687307843 − 20701491148800000n25 (420n5 − 210n4 + 7n2 − 2) 1 1 1 11 79 1 <1+ + 2+ − + + 3 4 5 2n 8n 240n 1920n 26880n 396n6 exp{r} < 1 + for n ≥ 3, which proves (9) for n ≥ 3. The case n = 2 is easily checked. 3 Final Remarks The first three terms of our new inequalities (1) for θn are the best possible asymptotically while the fourth term in the upper bound is subject to improvement. Proposition 4 and Proposition 6 are special cases of the general expansion of n!, with an error term, which can be proved by using the Euler–Maclaurin sum formula. However, our proofs are much more elementary, and can be extended to any degree of accuracy desired. Still, our proofs do not supply the general formula for the coefficients in the exponential version, although perhaps they can be properly modified to do so. Also, our technique for proving the positivity of certain large degree polynomials seems to argue for a general property of polynomials P (x) with real coefficients that are positive for x ≥ a. The property in question is that there exists a b ≥ a such that the quotient polynomial Q(x) in the division algorithm P (x) ≡ Q(x)(x − b) + R has all its coefficients positive. Finally, the inequalities (1) can, with more work, be extended to degrees three, four, etc., where the main coefficients are given by the formula of Karatsuba [3]. We also conjecture that the correction term θn is completely monotonic. Acknowledgements We thank the referee for the reference [1]. We would like to thank Daniel CamposSalas for some helpful suggestions about the concavity. MBV acknowledges support for the Vicerrectorı́a de Investigación of the University of Costa Rica. References [1] Horst Alzer, “On Ramanujan’s Double Inequality for the Gamma Function”, Bull. London Math. Soc. 35 (2003) 601-607. 7 [2] M. D. Hirschhorn, “A New Version of Stirling’s Formula”, Mathl. Gazette 90 (2006), 286–291. [3] E. A. Karatsuba, “On the asymptotic representation of the Euler gamma function by Ramanujan”. J. Comput. Appl. Math. 135 (2001), 225–240. [4] S. Ramanujan, The Lost Notebook and other Unpublished Papers, S. Raghavan and S. S. Rangachari, eds., Narosa, New Delhi, 1987. [5] M. B. Villarino, D. Campos-Salas and J. Carvajal-Rojas, “On the monotonicity of the correction term in Ramanujan’s factorial approximation”, Mathl. Gazette 97 (2013), to appear. 8