Ramanujan's Harmonic Number Expansion

2 downloads 0 Views 97KB Size Report
Nov 17, 2005 - to the harmonic series produces the general formula and error ... of Ramanujan's expansion, nor to a general formula for the coefficient of 1 mk .
arXiv:math/0511335v2 [math.CA] 17 Nov 2005

Ramanujan’s Harmonic Number Expansion Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica February 2, 2008 Abstract An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number.

1

Introduction

Entry 9 of Chapter 38 of B. Berndt’s edition of Ramanujan’s Notebooks, Volume 5 [1, p. 521] reads: “Let m := n X 1 k=1

k

n(n+1) , 2



where n is a positive integer. Then, as n approaches infinity,

1 1 1 1 1 1 ln(2m) + γ + − + − + 2 3 4 2 12m 120m 630m 1680m 2310m5



29 2833 140051 191 + − + − [· · · ].” 6 7 8 360360m 30030m 1166880m 17459442m9

Berndt’s proof simply verifies (as he himself explicitly notes) that Ramanujan’s expansion coincides with the standard Euler expansion Hn :=

n X 1 k=1

k

∼ ln n + γ + = ln n + γ −

1 1 1 − + − [· · · ] 2n 12n2 120n4 ∞ X Bk k=1

nk

where Bk denotes the k th Bernoulli number and γ := 0.57721 · · · is Euler’s constant. 1

However, Berndt does not give the general formula for the coefficient of m1k in Ramanujan’s expansion, nor does he prove that it is an asymptotic series in the sense that the error in the value obtained by stopping at any particular stage in Ramanujan’s series is less than the next term in the series. Indeed we have been unable to find any error analysis of Ramanujan’s series. Although it is true that the asymptotics of the harmonic numbers were already determined by the Euler expansion, later mathematicians have offered alternative approximative formulas (see [2, 3, 4, 5, 6, 8]). Ramanujan’s formula is one of the most accurate (see [8]).. However, the Euler expansion apparently does not easily lend itself to an error analysis of Ramanujan’s expansion, nor to a general formula for the coefficient of m1k . In an earlier paper (see [7]) we proved that the first five terms of the Ramanujan expansion indeed form an asymptotic expansion in the sense above, but we did not prove the general case. The general case is the subject of the following: Theorem 1. For any integer p > 1 define: (−1)p−1 Rp := 2p · 8p

(

1+

p   X p k=1

 ) 1 , (−4)k B2k k 2

(1)

where B2k (x) is the Bernoulli polynomial of order 2k. Put m :=

n(n + 1) 2

(2)

where n is a positive integer. Then, for every integer r > 1, there exists a Θr , 0 < Θr < 1, for which the following equation is true: r

X Rp 1 1 Rr+1 1 1 + Θr · r+1 1 + + + · · · + = ln(2m) + γ + p 2 3 n 2 m m p=1

(3)

We observe that the formula for Rp can be written symbolically as follows: 1 Rp = − 2p where we write B2m

2

1 2





4B 2 − 1 8

p

(4)

in place of B 2m after carrying out the above expansion.

Proof of Ramanujan’s Expansion

Proof. We begin with the half-integer approximation to Hn due to DeTemple and Wang (see [4]): For any positive integer r there exists a θr , 0 < θr < 1, for which the following 2

equation is true: 

1 Hn = ln n + 2



+γ+

r X

Dp n+

p=1

where

 1 2p 2

+ θr ·

Dr+1 2r+2 n + 21

 B2p 21 Dp := − . 2p Since 

we obtain: r X p=1

Dp n+

 1 2p 2

r X

1 n+ 2

2

= 2m +

(5)

(6) 1 4

Dp  1 p 1 + 8m p=1 −p  r X 1 Dp 1+ = (2m)p 8m p=1  ∞  r X 1 Dp X −p = p k (2m) k=0 k 8 mk p=1   r ∞ X Dp X 1 1 k k+p−1 = (−1) · p+k p k k 2 k=0 8 m p=1 ) (   p−1 r X X 1 p − 1 1 Ds p−s (−1) · + Er = s p−s p p − s 2 8 m p=1 s=0 =

(2m)p

where     ∞ ∞ D1 X 1 1 1 D2 X 1 k k k k+1 Er := 1 (−1) (−1) · 1+k + 2 · 2+k + · · · k k k 8 m k 2 k=r 2 k=r−1 8 m   ∞ 1 Dr X 1 k k+r−1 (−1) + r · r+k k k 2 k=1 8 m Substituting the right hand side of the last equation into the right hand side of (5) we obtain: ( p−1 )     r X X Ds 1 1 1 Dr+1 p − 1 +γ+ · p + Er + θr · Hn = ln n + (−1)p−s 2r+2 . s p−s p−s 8 2 2 m n + 21 p=1 s=0 (7) 3

Moreover, 2   ln n + 21 1 = ln n + 2 2   1 1 = ln 2m + 2 4   1 1 1 = ln(2m) + ln 1 + 2 2 8m ∞ 1 1 1X (−1)l−1 l l = ln(2m) + 2 2 l=1 l8 m r

=

1 1X 1 (−1)l−1 l l + ǫr ln(2m) + 2 2 l=1 l8 m

where ǫr :=

∞ X

(−1)l−1

l=r+1

1 . 2l8l ml

Substituting the right-hand side of this last equation into (7) we obtain ( p−1 )   r r X X Ds 1 1 p − 1 1 1X 1 (−1)l−1 l l + γ + Hn = ln(2m) + (−1)p−s · p s p−s p−s 8 2 2 l8 m 2 m p=1 s=0 l=1

Dr+1 2r+2 ] n + 21 ( )   p−1 r X X 1 1 1 p − 1 1 D s p−s = ln(2m) + γ + (−1)p−1 + (−1) · 2 2p8p s=0 2s p − s 8p−s mp p=1 + ǫr + Er + θr ·

+ ǫr + Er + θr ·

Dr+1 2r+2 n + 21

Therefore, we have obtained Ramanujan’s expansion into powers of cient of m1p is Rp =

(

)   p−1 X 1 p − 1 D 1 s + (−1)p−s (−1)p−1 p − s 8p−s 2p8p s=0 2s

4

1 , m

and the coeffi-

(8)

But,     B2s ( 12 ) 1 1 p − 1 Ds p−s p−s p − 1 2s (−1) = − s (−1) s p−s p−s p−s 8 p−s 8 2 2   1  p−1 1 p−s−1 B2s 2 = (−1) 2s2s p − s 8p−s and therefore   p−1 X 1 Ds 1 p−s p − 1 + (−1) Rp = (−1) p s p−s p−s 8 2p8 2 s=0   p−1 1  X B p−1 1 1 2s p−s−1 p−1 2 (−1) + = (−1) p − s 8p−s 2p8p s=0 2s2s ( )   p 1  X B 1 1 p − 1 2s 2 = (−1)p−1 (−1)s + p − s 8p−s 2p8p s=1 2s2s ) (    p 1 X B 1 p 1 1 2s 2 (−1)s + · = (−1)p−1 2p8p s=1 2 · 2s p s 8p−s (  ) p   X (−1)p−1 1 p = (−4)s B2s 1+ p s 2p8 2 s=1 p−1

Thus. the formula for Hn takes the form: r

X (−1)p−1 1 Hn = ln(2m) + γ + 2 2p8p p=1 + ǫr + Er + θr ·

 ) p   X 1 1 p s (−4) B2s · p 1+ s 2 m s=1

(

Dr+1 2r+2 n + 12

(9) (10)

We see that (9) is the Ramanujan expansion with the general formula for the coefficient, Rp , of m1p , as given in the statement of the theorem, while (10) is (an undeveloped form of) the error term. We will now estimate the error. To do so we will use the fact that the sum of a convergent alternating series, whose terms (taken with positive sign) decrease monotonically to zero, is equal to any partial sum plus a positive proper fraction of the first neglected term (with sign). Thus, ∞ X 1 1 (−1)l−1 l l = αr (−1)r ǫr := 2l8 m 2(r + 1)8(r+1) mr+1 l=r+1 5

where 0 < αr < 1. Moreover,     ∞ ∞ 1 1 1 D2 X 1 D1 X k k k k+1 (−1) (−1) · 1+k + 2 · 2+k + · · · Er = 1 k k k 8 m k 2 k=r 2 k=r−1 8 m   ∞ 1 Dr X 1 k k+r−1 + r (−1) · r+k k k 2 8 m k=1         D1 1 1 1 r D1 Dr 1 r r r−1 1 r = δ1 1 (−1) + δ2 2 (−1) + · · · + δr r (−1) r r−1 1 r 8 r−1 8 1 8 2 2 2 mr+1         D1 1 r 1 D2 r Dr r 1 = ∆r (−1)r + 2 (−1)r−1 + · · · + r (−1)1 1 r r−1 r 8 r−1 8 1 81 2 2 2 where 0 < δk < 1 for k = 1, 2, · · · , r and 0 < ∆r < 1. Finally θr ·

Dr+1 Dr+1 = θr ·  1 2r+2 n+ 2 (2m)r+1 1 +

where0 < δr+1 < 1. Thus, the total error is equal to ǫr + Er + θr · =Θr ·

(

 1 r+1 8m

= δr+1 ·

Dr+1 1 · r+1 r+1 2 m

Dr+1 2r+2 n + 21

)   r+1 X 1 r D 1 1 2q r−q+1 + (−1) (−1)r r − q + 1 8r−q+1 mr+1 2(r + 1)8(r+1) q=1 2q

=Θr · Rr+1

by (8), where 0 < Θr < 1, which is of the form as claimed in the theorem. This completes the proof.

Acknowledgment Support from the Vicerrector´ıa de Investigaci´on of the University of Costa Rica is acknowledged.

References [1] B. Berndt, Ramanujan’s Notebooks, Volume 5, Springer, New York, 1998. [2] E. Ces`aro, “Sur la serie harmonique”, Nouv. Ann. (3) IV (1885), 295–296. 6

[3] Ch.-P. Chen and F. Qi, The best bounds of the n-th harmonic number, Global Journal of Mathematics and Mathematical Sciences 2 (2006), accepted. The best lower and upper bounds of harmonic sequence, RGMIA Research Report Collection 6 (2003), no. 2, Article 14. The best bounds of harmonic sequence, available online at http://front.math.ucdavis.edu/math.CA/0306233. [4] D. DeTemple and S-H Wang “Half-integer Approximations for the Partial Sums of the Harmonic Series” Journal of Mathematical Analysis and Applications, 160 (1991), 149156. [5] A. Lodge, “An approximate expression for the value of 1 + of Mathematics 30 (1904), 103–107.

1 1 1 + + · · · + ,” Messenger 2 3 r

[6] L. T´oth, and S. Mare “E 3432” American Mathematical Monthly, 98 (1991), no 3, 264. [7] M. Villarino, “Ramanujan’s Approximation to the nth Partial Sum of the Harmonic Series”, preprint, arXiv.math.CA/0402354 [8] M. Villarino, “Best arXiv.math.CA/0510585

Bounds

for

7

the

Harmonic

Numbers”,

preprint,