Pi, Euler Numbers, and Asymptotic Expansions
Author(s): J. M. Borwein, P. B. Borwein and K. Dilcher
Source: The American Mathematical Monthly, Vol. 96, No. 8 (Oct., 1989), pp. 681-687
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2324715
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Pi, Euler Numbers,and Asymptotic
Expansions
J.M. BORWEIN,P. B. BORWEIN, andK. DILCHER1,Dalhousie University,
Halifax,Canada
M. BORWEIN
JONATHAN
was an OntarioRhodesScholar(1971)at Jesus
wherehe completed
a D. Phil.(1974)withMichaelDempCollege,Oxford,
ster.Since1974he hasworked
at DalhousieUniversity
whereheis professor
of mathematics.
He has alsobeenon faculty
at Carnegie-Mellon
University
(1980-82).He was the1987Coxeter-James
lecturer
oftheCanadianMathematicalSocietyand was awardedtheAtlanticProvinces
Councilon the
Sciences1988 Gold Medal forResearch.His researchinterests
include
functional
classicalanalysis,
andoptimization
analysis,
theory.
PETER B. BORWEIN
of
obtaineda Ph.D. (1979) fromthe University
British
underthesupervision
ofDavidBoyd.He spent1979-80as
Columbia,
a NATO researchfellowin Oxford.Since thenhe has been on facultyat
9
Dalhousie(exceptfora sabbatical
ofToronto)andis
yearat theUniversity
of Mathematics.
now AssociateProfessor
His researchinterests
include
classicalanalysis,
andcomplexity
theory,
approximation
theory.
DILCHER received
hisundergraduate
KARuL
education
and a Dipl. Math.
in WestGermany.
Clausthal
degreeat Technische
Universitat
He completed
his Ph.D (1983) at Queen'sUniversity
in Kingston,
OntariowithPaulo
Ribenboim.
Since1984he has beenteaching
at Dalhousie,wherehe is now
AssistantProfessor.
His research
interests
includeBernoulli
numbers
and
and classicalcomplex
polynomials,
analysis.
at 500,000terms,
seriesfor'T, truncated
1. Introduction.
Gregory's
givesto forty
places
500,000 (1)k-1
4 E
k=1
2k
1 = 3.141590653589793240462643383269502884197.
The numberon therightis not ir to forty
places.As one wouldexpect,the6th
is
is thatthenext10 digitsare
the
The
digitafter decimalpoint wrong.
surprise
In
4
aren't
correct.
Thisintriguing
observathe
underlined
correct. fact,only
digits
us
R.
D.
North
of
Colorado
with
a
was
sent
to
tion
by
[10]
Springs
requestforan
article
is
to
that
Two related
The
of
this
explanation. point
provide
explanation.
in partbyNSERC ofCanada.
'Researchof theauthors
supported
681
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682
[October
J. M. BORWEIN, P. B. BORWEIN, AND K. DILCHER
examples,to fifty
digits,
are
50,000 (W)k-1
IT
2
2k - 1
k-i
1.5707863267948976192313211916397520520985833147388
-
1
and
log2-=
=
-1
5
- 61
k+1
50,000 (1)
k
k=1
.69313718065994530939723212147417656804830013446572,
-1
1
- 16
2
272
The numbers
undertheunderlined
whereall but theunderlined
digitsare correct.
thatmustbe addedto correct
these.The numbers
digitsare thenumbers
1, - 1, 5,
- 61 are thefirstfourEulernumbers
while1, - 1, 2, - 16, 272 are thefirstfive
thesesequences
of generating
consisted
Ourprocessof discovery
numbers.
tangent
themwiththeaid of Sloane'sHandbookofIntegerSequences
and thenidentifying
oftheerrorin
in eachcase,is an asymptotic
[11].Whatone is observing,
expansion
of theexpansionare
The amusingdetailis thatthecoefficients
Eulersummation.
All of thisis explained
integers.
byTheorem1.
numbers
{ E1}, thetangent
The standardfactswe needabouttheEulernumbers
{ T }, and theBernoulli
{ B1}, mayall be foundin [1]orin [6].Thenumbers
numbers
of thepowerseries
are definedas thecoefficients
n E2n 2n
??
secz = E (-1)
(2)(1.1)
2n)! 2+
(0n+
tanz
=
E (
1)
0B
E
Zn
n=O
z
e
z
1
n=
2
(2 n+1)!
~
and To
ad01
1,
n
(1.2)
12
~~~~~~~~~~~~~(
n!
Theysatisfy
therelations
E
=
(2kE2k
E2n+1 = 0,
0,
Bn= 2n(2n"-1)
n>1
(1.4)
15
and
af
(k
?)
Bk=
(1.6)
of { En}, {Tn}, and {Bn}. The
Thesethreeidentities
allowfortheeasygeneration
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1989]
683
PI, EULER NUMBERS, AND ASYMPTOTIC EXPANSIONS
firstfewvaluesarerecorded
below.
n
0
n~
T7
1
Bn,
1
1
-
2
0
-
1
0
3
5
4
6
00
7
-61013
2
-16
8
165
0
272
0
0
0
It is clearfrom(1.4) thattheEulernumbers
are integral.
From(1.5) and (1.6) it
followsthatthetangent
areintegers.
numbers
Also,
-
_E_n_12n__
I2nI
2(2n)!
ad_
ad
.72n?1
( v2
I2nl
as followsfrom(5.1) and(5.2) below.Themaincontent
ofthisnoteis thefollowing
theorem.
The simpleproofwe offer
relieson theBooleSummation
which
Formula,
is a prettybut less well-known
The detailsare
analogueof Euler summation.
containedin Sections2 and3 (exceptforc] whichis a straightforward
of
application
Eulersummation).
Morecomplicated
can be baseddirectly
on Euler
developments
summation
or on resultsin [9].
THEOREM1. Thefollowing
asymptotic
expansionshold:
-
a]
2
2~ E_
2k -i
k-i
N/2 (_1)
b] log2- Y,
k=1
=0 N
k1
k
k
2m?1
1
1
5
61
-N
NT3
N5
V
N7
1
co
- N +
m=1
Trn-i
=N
2
1
1
2
-N
N2
N4
16
272
N
N
and
c]
+
k 22N2
6kF' k=1
m=O N2m+1
1
2N
N
1
6N3
30N'
42N7
Fromtheasymptotics
of { E~,} and { Bn} and (1.5) we see thateachoftheabove
infinite
seriesis everywhere
thecorrect
oftheirasymptotics
divergent;
interpretation
is
K
((2K + 1)!
00E
E
a']
~~m=1
N2
00
T_
b']
L~M=N 2m
c']
f~0NB2m1
m=1N
-
rn- N2
K
((2K
T_
+ 0
m=
K
(E7rN)2K
B2
m_1 N2m1
+
+ 1)!
(ElTN)2K-1
(2K + 1)!\
(2u,gN)2K?1l
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[October
J. M. BORWEIN, P. B. BORWEIN, AND K. DILCHER
684
ofN
concealedbytheordersymbolis independent
wherein eachcase theconstant
10 worksin all cases.
and K. In fact,theconstant
2. The Boole SummationFormula.The Euler polynomialsEn(x) can be defined
function
by thegenerating
-
et
tn
=
2etx
=
(Iti
n <
En(X))-
(2.1)
T);
1.
ofdegreen withleadingcoefficient
(see [1,p. 804]).Each En(x) is a polynomial
by
We also definetheperiodicEulerfunction
En(x)
En(X + 1) = -En(X)
forall x, and
En(x) = En(x)
for 0 < x < 1.
derivatives
up to the(n - I)st order.
It can be shownthatEn(x) has continuous
The following is known as Boole's summationformula(see, for example,
[9, p. 34]).
LEMMA 1. Let f(t) be a functionwithm continuousderivatives,definedon the
intervalx < t < x + w. Thenfor 0 < h < 1
f(x + hw)
m-1
=
1
W;k
-Ek(h)
klOk
k=O
-
- (f (k)(X + W) + f
(k)(X))
2
+ Rm,
where
Mj (E
RmI=
(h
t)
(m)(x
+
wt)dt
bypartsof
byrepeatedintegration
is easyto establish
formula
This summation
to Euler,
was
known
in
that
this
formula
is
remarked
It
26]
[9,
p.
above
the
integral.
term.Also notethatLemma1 turns
theremainder
forpolynomialf and without
remainder
termifwe replaceh by h/l and
withLagrange's
intoTaylor'sformula
let coapproachzero.
we havein mind,
ofLemma1 fortheapplications
version
To derivea convenient
=
w
on
f.
restrictions
1
further
and
impose
we set
derivatives,definedon t > x.
LEMMA 2. Let f be a functionwithm continuous
0 as t ooforall k = 0,1, ..., m. Then for 0 < h < 1
Suppose thatf(k)(t)
+ h + v)
(1)f(x
v=O
rn-Ek(h)f(k)(X)
k=O
+2R
where
R m
= _|
2
(h
)(M)(x
f
(M - ~1)!
+ t)dt.
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1989]
PI, EULER NUMBERS, AND ASYMPTOTIC EXPANSIONS
685
3. The Remainderfor Gregory'sSeries. The Euler numbersEn may also be
function
definedby thegenerating
2 _
et + et
0 tn
n!(31
n=
(3.1) with(2.1),we see that
Comparing
En= 2nEn(
.
(3.2)
explainedby thenext
is entirely
in theintroduction
mentioned
The phenomenon
proposition-ifwe set n = 500,000.It is also clearthatwe willgetsimilarpatterns
m.
forn = 10/2 withanypositiveinteger
PROPOSITION 1. For positive integersn and M we have
k
oo (_)
4
k= 2k + 1
where
-
I
2E2
(_I)nE
%(M) ~
PI
Proof. Apply Lemma 2 withf(x)
00
M
.
k=O
-
v=0 n + v + 1/2
~~~~~~~(3.3)
(M),
21E2MI
I ___z
(2n)2M?l
=
l/x; thenset x = n and h = 1/2. We get
rn-i E1(1/2) (_l)
(1)v
+ R
(2n )2k?l +
k-O
2k!
kk!+R
nk?l
m
(3.4)
with
Rm
m=
I
ooEmi_(h- t) (-1) m!dt
dt.
(x + t)m+l
2 JO (mr-1)!
side is seento be
We multiply
bothsidesof (3.4) by 2(-1)n. Thentheleft-hand
sideof (3.3). Afterreplacingm by2M + 1 and taking
identicalwiththeleft-hand
vanish,we see thatthe
Eulernumbers
intoaccount(3.2) andthefactthatodd-index
sidesof (3.3) and (3.4) agree.To estimatetheerror
firsttermson theright-hand
inequality,
term,we use thefollowing
IE2M(x) I < 2 2MIE2MI
forO < x < 1
nowleadsto theerrorestimate
out theintegration
(see, e.g.,[1, p. 805]).Carrying
0
1.
givenin Proposition
4. AnAnalogueForlog2. Lemma2 can also be usedto derivea resultsimilarto
of theseries
truncations
1, concerning
Proposition
log2= E
k=1
(
1)~
k
.
~
~
~~~~(41
(4.1)
1. It
In thiscase thetangent
numbers
TnwillplaytheroleoftheEn in Proposition
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[October
J.M. BORWEIN,P. B. BORWEIN,AND K. DILCHER
686
followsfromthe identity
1
tanz=T
ez2iz+1
2e21z
1)
togetherwith(1.2) and (2.1) that
as in [9, p. 28]. The
(4.2)
2nEn(1)
Tn-(-1)
Tncan be computedusing the recurrencerelationTo = 1 and
k=
k)2 T-k + Tn= 0 forn > 1.
Otherpropertiescan be found,e.g.,in [81or [9, Ch. 21.
n and M we have
PROPOSITION 2. Forpositiveintegers
= (-1)
(
E
k=n+l1
n(1 2
2
+
E (22k
k==1 (2n)
) + R2(M),
(4.3)
k)
where
JR
AR2(M)
I<
J~E2MI
(2n)2M+l
Proof. We proceed as in the proof of Proposition1. Here we take x = n and
h = 1. Using (4.2) and the fact that To = 1 and T2k = 0 for k > 1, we get the
side of (4.3). The remaindertermis estimatedas in the
summationon theright-hand
0
proofof Proposition1.
of
Using Proposition2 withn = 10m/2one again getsmanymorecorrectdigits
log 2 than is suggestedby the errortermof the Taylor series.
5. Generalizations.Proposition1 and 2 can be extendedeasily in two different
directions.
i). The well-knowninfiniteseries(see, e.g., [1, p. 8071)
kO
k=O
(_)k+
)
(2nI
=
(2k + 1
2)
__2_
= (1 -21-2n)(2n)
s.2n+1
(n
=
0,1,
(5.1)
**.),
and
oo
k=1
= (22n1
-
1) IB2n7I 2n
(n = 1,2,
)
(5.2)
can be consideredas extensionsof Gregory'sseriesand of (4.1). These seriesadmit
exact analogues to Propositions1 and 2; one only has to replace f(x) = l/x by
x-(2n), in theproofs.
f(x) = x- (2n+ ) respectively
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1989]
PI, EULER NUMBERS, AND ASYMPTOTIC EXPANSIONS
687
for
results
leadsto similar
formula
summation
We notethattheEuler-MacLaurin
00
k=1
IB2n122n-1
(2n)!
'
(5)
numbers
of theBernoulli
wheremultiples
B2n taketheplace of theEn and Tnin
2.
and
1
Propositions
was
formulas
and Boolesummation
oftheEuler-MacLaurin
ii). A generalization
series
the
of
to
character
analogues
This
can
be
applied
Berndt
[3].
derivedby
1 and 2 are thenplayedby
(5.1)-(5.3). The rolesof theEn and Tnin Proposition
or byrelatednumbers.
numbers
Bernoulli
generalized
results
observedin theintroduction
The phenomenon
Comments.
6. Additional
fromtakingN to be a powerof ten; takingN = 2 * io0 also leads to "clean"
[1],[5],[6],and [9] includethebasicmaterialon Bernoulli
References
expressions.
withtheircalculation,and [2]
and Euler numbers,while[8] deals extensively
of
Muchon thecalculation
analogueof Pascal'striangle.
describesan entertaining
is treatedin [5],[6],
maybe foundin [4]. Eulersummation
pi and relatedmatters
on thecomputain [9].Relatedmaterial
is treated
and [9],whileBoole summation
seriesis givenin [7].
ofalternating
tionand acceleration
was observedby M. R. Powell
Addedin Proof. A versionof thephenomeon
Gazette,66 (1982)
(see The Mathematical
wereoffered
and variousexplanations
220-221,and 67(1983)171-188).
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