An Antisymmetric Formula for Euler`s Constant

An Antisymmetric Formula for Euler's Constant
Author(s): Jonathan Sondow
Reviewed work(s):
Source: Mathematics Magazine, Vol. 71, No. 3 (Jun., 1998), pp. 219-220
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2691211 .
Accessed: 25/12/2011 10:51
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to
Mathematics Magazine.
http://www.jstor.org
219
VOL. 71, NO. 3, JUNE 1998
ProofWithoutWords: BijectionBetweenCertainLatticePaths
Dedicated to Ernst Specker on the occasion of his 78th birthday.
fliphorizontally
/-
-
-
\--
-
-
-
-
-
-
-
-
-
-
-
0
-
-
-
-
-
-
-
-
-
-
p
ab: lattice path with startingpoint and endpoint on the same (given) level
p: firstminimum on the path ab
pc: lattice path stayingabove initial level (non-ruin path)
Conclusion
There exist as many non-ruinpaths of length 2n as paths of length 2n
with startingpoint and endpoint on the same level, namely (
).
-NORBERT
HUNGERBUHLER
ETH-ZENTRUM
CH-8092 ZURICH
SW,NITZERLAND
An Antisymmetric
FormulaforEuler'sConstant
JONATHAN SONDOW
209 West 97thStreet
New York,NY 10025
The formula
1
lim+
(
X)
showsthatEuler's constant,y,whichis defined(see [1]) by
1
1
+ -+
y= im 1 +
-logn
fl
2
-*00
(2)
is thelimitas x approaches1 fromabove ofa serieswhosetermsare antisymmetric
in
1+ of the difference
n and x. The formulaalso impliesthat y is the limitas x
betweenthe p-seriesZ7= 1/nx and the geometricseriesEcI=, 1/x", because
x
n=I n
=1 x
for x > 1. On the otherhand, since the geometricseries sums to 1/(x - 1), the
formulais itselfan immediateconsequenceof the fact(see [4, Section2.1]) that
n
limr(x)
where; (x) =
xt
ftI=
- 1f =
x
zetafuLnction.
betweeny
(For a connection
/n is theRiemann
MATHEMATICS MAGAZINE
220
and
and the zeros of the zeta function,as well as a wealth of otherinformation
on y, see [2].)
references
We now givean independentproofof the formula.First,note that
/11I
y= lim
E
(
T
n--oo k=1
+1)
of y, because
is equivalentto the definition
lim (logn
-
log('n + 1))
+ 1)
lim log(
=
=
.
Now write
1
0, [+cit
Cdt.
and
+
jk?dt
k
~?dt
log( n+ 1)
+
as sumsofintegrals.It followsthatthelimitsin equations(1) and (2) can be writtenas
11f--f
'it
lim
+
(n-'I
x1
(3)
'
and
(1
jn?ldt)
The two limitsare thereforethe same, since the latterseries is the
respectively.
so
limitof the formerseries,whichwe now showconvergesuniformly,
term-by-term
the limit and the summationis justified.To prove uniform
that interchanging
convergenceoftheseriesin formula(3) on theinterval[1,2], we applytheWeierstrass
M-test(see, e.g., [3]), usingthe seriesEnr-2 forcomparison:
0<
-
t'
< xhlx1f|?(fcl)
fI
(
(
dct=
nx
dt
tA)
___*n
1 <
f?
(J|
X1dUt)
dt
n
formulafor Euler's
for 1 < x < 2. This completesthe proof of the antisymmetric
constant.
REFERENCES
1. JohnV. Baxley,Euler's constant,Taylor'sformula,and slowlyconvergingseries,this MAGAZINE 65
(1992), 302-313.
2. Jeffrey
Nunemacher,On computingEuler's constant,thisMAGAZINE 65 (1992), 313-322.
3. WalterRudin,Principlesof Mathematical
Analysis,3rd ed., McGraw-Hill,New York,NY, 1976.
Press,Londonand New
4. E. C. Titchmarsh,
The TheoryoftheRienlannZeta-Futnction,
OxfordUniversity
York,1951.