SUMMATION A!D TABLE OF FI1ITE SUMS
by
ROBERT DELMER STALLE!
A THESIS
subnitted to
OREGON STATE COLlEGE
in partial fulfillment of
the requirementh for the
degree of
MASTER OF ARTS
June l94
APPROVED:
Professor of Mathematics In Charge of Major
Head of
Deparent
of Mathematics
Chairman of School Graduate Committee
Dean of the Graduate School
ACKOEDGE!'T
The writer
dshes to eicpreßs his thanks
to Dr. W. E. Mime, Head of the Department of
Mathenatics, who has been
a
constant guide and
inspiration in the writing of this thesis.
TABLE OF CONTENTS
I.
i
Finite calculus analogous to infinitesimal
calculus. . .. . .. . . . . . . . . . . . . ..
. ..
e s
2
Theconstantofsuirrnation.........................
2
3
a
Suniming
as
the
inverse of
. . . . . . . . .
perfornungA............
31nite calculus as a brancn of niathematics........
Application of finite 5lflITh1tiOfl...................
II.
III
5
LVELOPMENT OF SULTION FORiRLAS....................
6
ttethods...........................a..........,....
Three genera]. sum formulas........................
6
6
S1ThATION FORMULAS DERIVED B
OF A
Z
INVERSION
FELkTION....,..................,...........
TIlE
7
s urnmation by parts..................15......
Ratlona]. functions................................
7
Gamma and related functions........,...........
9
...
Ecponential and logarithrnic functions...... .
Thigonoretric arÎ hyperbolic functons..........,.
Combinations of elementary functions......,.....
IV.
4
SUMUATION BY IfTHODS OF APPDXIMATION..............,.
Tewton s formula . . . . a a a a S a C .
a e a
a
.
a
Extensionofpartialsunmation................a...
Formulas relating a sum to an ifltegral..a.aaaaaaa.
Sumfromeverym'thterm........aa..a..aaa........
a
s
e
a a
a a
a
J-3
14
15
15
15
16
17
V.
TABLE OFST.Thß,..,,..,,...,.,,.....,....,,,...........
18
VI.
SLThMTION OF A SPECIAL TYPE OF
26
VI I
BIBLIOGRAPHY.
. a
a a a
a a a a a a
.
.
POER SERIES..........
a . .
a a a a s . . . . . .
e . * . a a a .
44
TABlE OF FINITE
STThSIAATION AND
I
To
T}OR!
STJ
OF FINITE SU1ATION
the knowledge of' the writer,
no
table
of'
finite
sum formulas
of value as a reference is in existence. For this reason a table
of such formulas, analogous to a table of' integral formulas, is
presented here. The formulas for this table were selected with
their utility in mind.
A
series.
finite
sum
formula
is
a
Finite suntion is the
relation giving the
eluation
of
a
sum
of a finite
finite series.
This valuation cari be accomplished by the development and application
of general
sum
formulas.
These formulas are of closed form and give
the exact or approximate sum, depending on the type of
series.
The theory of finite sunvation is presented first and is
followed by the development of
and
finite sun formulas, both general
special. Finally the table is presented.
As we will see, the subject of summation arises as a branch
of the calculus of finite differences.
finite summation will be
Throughout
throwgh
this paper
Uence
this treatment of
this calculus.
numbers enclosed
to relationships in the discussion, whereas
parentheses refer to formulas in the table.
in parentheses refer
nwbers not enclosed in
2
Finite calculus analoous to infinitesiiial calculus.
There is a striking analogy bet'reen the finite calculus, or
calculus of finite differences, and the infinitesimal calculus.
Consider the function Ut subjected to these calculi.
The two
fundanntal operations of the infinitesimal calculus are
differentiation,
L
dt
and its
inrse,
lim
h40
%th'1t
h
The two fundanental operations of
integration.
¿ - operation,
the finite calculus aro the
LUt
UUt
h
and its inverse.
The analogy is
nt
perfect because of the
additional Uniting operation in differentiation.
Results in finite
calculus may be reduced to the analogous results in infinitesimal
calculus at any time if we perform this additional limiting operation.
In fact by stxiying finite calculus
gain a greater perspective
onto and insight into infinitesimal calculus.
In the finite calculus we may assume, without loss of generality,
that the linear subatitution, t = hx, haa been made
Lt * bx,
L
X is unity.
Hence ¿ performed on u
Now, since
nw
gives:
(1)
SuDud.n2 as the inverse of performing ¿
Let us inveitigate the inverse of the
¿\
operatIon, which
3
It is represented by
George i3oole cafled difference integration.
¿.
the srnbo1
to n,
in
we
u,
if
A 1
then
=
from
Given
exists such that
a function
A
=
If we SUYa
'-a
for values of x
obtain:
n
'n-i.].
'-uX =VI
Xl
(2)
.
m
mus
i iç
term
is
The
the series whose general
inay be found we niay sun
u.
constant of suiiniiation.
is interrogative and hence that
Note that the operation
we have no assurance of the existence or uniqueness of the result
u.
from perforning it on a function
Consider a case in
is not guaranteed.
of
-1
and
V+
an expression for
results from
¿
exists.
which
reesent
and
Hence we let V
the uniqueness
Now
be an expression
Now
all such expressions.
for
to Lind
thereby the relation between possible
we consider the
Removing the parentheses we see
relation, £v
¿1&)
=
=
Thus the
O.
expression for
(1, p.79-3l).
is a general periodic function of period one
In
finite calculua x assimies successive values differing by unity so
we may consider
the rrre
as a constant.
suggestive notation
I
for
Thus
Ô1
=
v+
c.
We use
and have;
:=v+c.
(3)
4
If
we
consider this
sum from
m
to n
have by (2):
we
n+l
n+l
n
U=Vx+C
(4)
Finite calculus as a branch of mathematics.
A look at finite calculus as a subject is In order.
is
some
There
literature about the technical terms used
to the close analor with infinitesimal calculus
confusion in
Due
(1, p.&L-E2).
many terna are borrowed here.
Below is a list of terms from finite
calculus paired with the analogous terms of infinitesimal calculus.
From this and previous discussion their meaning is held to be
self evident.
Finite
calculus
Performing
Infinitesimal calculus
Differentiating
¿\
Difference integral
Indefinite integral
Sum
Definite integral
Difference calculus
Differential calculus
Sum calculus
Integral calculus
Suinand
Inte grand
Infinitesima]. calculus includes evaluation of
by other methods
than through the
indefinite
definite integrals
integral.
it is convenient to include in finite calculus
Likewise
summation performed
by other methods than through the difference integral.
Sonatines
it is necessary to define and tabulate a new function in order to
5
perform
exact
evaluation as for
X
the integral,
dx =
:
and the sum,
=f(n) -fm-1).
Sometimes approximate
evaluation
is the beet we can do as in the
case of the integral,
_____
(lT
dx,
)Vsinx
o
and the sum,
n
arc tan x.
i
Summation
of infinite series is analogous to evaluation of a
integral with an infinite limit.
is also
definite
Hence sunmation of infinite series
included in the calculus of
finite differences.
Applications of finite summation.
Some of the
methods to be discussed in the paper permit the
summation of convergent infinite series
By definition, the sum of the convergent
in closed form (3, p.18-23).
infinite series
n
is
fl-4
E
a
j:;].
a
j
Thus to sum the infinite serles is to evaluate
the sum of the first n terms and then the limit of this sum.
Another application of the sums and swmmation theory here
developed is in the solution of difference equations.
We proceed to develop these methods whereby we nay sum anr
finite series.
6
SU!ATIO'
T OF
DE VEI
II
ORULA3
Methods.
The two nx,st connon methods for derivation of finite sum
formulas are
(1) the
:
invorson of
iie of the close relation
methods
inc1e
been
a difference relation, and (2)
a sum and an integral.
Other
mathematical induction, operational attacks,
resolution of difference equations, use of generating functions,
use of geometric considerations, and use of the calculus of
probability
(4, p.109-141).
Part III includes formulas, with one exception exact, developed
by invarsion of a
formulas,
A
relation.
rart iv inclixles approximating sum
hich are derivad by the principle of inversion or by
using the sum-integral relation.
In the table, sums appear
5ii
form (4) with the limits
substituted, or in the easier written by equivalent form (3).
Three
enera1 sum formulas.
The derivations of formulas 2, 3, and 5 of the table come
under a separate heading.
In each case tho sums are expanded and
the identity of the expansions noted (6,
called Dirichlets ' sum-formula.
functions u
and V
p.9l3).
Formula 5 is
It is useful when one of the
can be suimmed exactly, but the other not.
7
III.
-
by inversion le to find
To sum u
and apply
31ThATION TORW1JIAS DERIVED BY THE
INVERSION OF A A RELATION
(3). For example from
that
cx = c e have fornula 1,
¿
L- relation uay be
In fact any
e = cx + C.
such
V
inverted to
ve
a sum.
SummatIon by
The
by
parts.
general formula for summation by parts, 4 is developed
this inversion.
?(e
have:
V
whence by summing each
ffrst
side from
in
to n:
n
n
lin
in
in
vixl
first
sum and
(5)
Lux Vx+i,
+
_tn+l
x
Solving for the
=
substituting EV= V1
tue
we hove
form of 5,
tni-1
n
=ux :vxi
L-xx
Im
in
I
Mjuctment of the limits and
second form.
The
reapplications of
from the second
auxmnand
n
_2ILu:v+l.
in
in the
is developed
third
form
it
its right.hand
to
in the
same manner.
analogous to integration by
parts.
first
form yLelds the
from the
sum.
The
first
by
r
=
i
fom'th is derived
Summation by
parts plays
a
role
C
tional fwictiona,
is defined for positive
The factorial (ax + b)
(ax
for negative
a)...(ax + b
+ b)(ax + b
ni:
am + a),
ni:
i
(6)
(ax + b)(ax + b + a)...(ax + b + ma - a)'
and for ni = O as unity.
If we invert:
+ b)(m+1) =
and divide through by
(ni
+ l)a(ax + b)(m),
+ 1), we obtain
a(rn
foririula 6:
(ax
Z(ax +b)
(7)
a(m+l)
+C.
Evidently ni cannot equal -1.
Formula 7
is
Aux+
seen to hold
if
u,cui+ ...
+
and reduce the resulting,
equating
like
we expand
in
as:
u u
A
ni-1 X
fractions.
oowers of x
4)(x)
The
4(x) and
A
's
may be determined by
this expansion.
The sums
resulting from application of 7 may be summed by 6.
Formula C is proved by use of the relation:
X
x\
(a)
al
for the binomial coefficient.
z 'a
(a)
ai
()
'
Hence,
(a+i)
(a
+ l)
+C=ai
Evidently a and x must be positive integers such that O<ax.
Q
Formula 9 used with 6 enables us to sun the power function
x11
(4, p.16E3-173).
We
have as the expansion of x'' by Ne'wton'a
formula, (23):
(m)
()
x0
Stirling's numbers of the second kind are defined as:
G:
arìd
hence we have formula 9.
relation
and
['J,
=
(10)
From (10) we have the recursion
initial condition:
m-1
m
Gn +mGn
Gn+l
n
O
=
1,
O
A brief table of
G
(11)
i
G
= O for n
O.
O
included later.
s is
Gamma and related functions.
The gamma function r(x) is the accepted generalization of
the factorial (x - 1)1
For
x>0,
r(x) =
Integration by parts yields
QJ ?1
the recursion
xr(x) = r(x +
which defines
r()
may be found in
for x
almost any
0.
dy.
(12)
relation:
1),
(13)
The foflowing fundamental relations
treatrient of
r():
lo
r__'-
i+th+
[
log f(x +
The
3-
2c2 -
X
-
5l?
571
{log ni + (x +
1) =
successive
+
24832
derivatis
of
l)log
.
...]
-
n
lor(x)
(34)
log(x
+v)J
.
(15)
are defined as the
triauxna, tetragamrna, et cetera functions and denoted
diganirna,
et cetera, respectively. These functions
are fundamental in the theory of sunirnation of rational fractions.
If a rational fraction s improper, division 4e1ds a rational
inteal function and
a proper
fraction.
Hence we need
consider
only rational integral functions and proper rational fractions.
A
by
rational integral function
formula 9 or
direct expression in factorials and use of 6 (5, p.27-23).
A proper rational fraction may be expressed as the sun of
fractions in
in
may be summed by
X and
the denominator is a povrer of a linear function
which
the numerator is a constant.
This
is affected by the
theory of rational fractions where the constant term in the linear
function .s a complex number.
'r
a) r
3.
(x
2 3-,
2, 3,
.., and a comnlex.
affected by use of the above
Essentially
wo
Hence our problem
gamma
and
are inverting a
sum
This sumnation
is
related functions.
relation but
attack
wi].]. be made on this si.vnrnation problem.
of the
digainma
function:
is to
From
a more
direct
the definition
11
f(x)
=1ogr(x
+1),
we have by (15):
r
n+2
=
(x)
L
n -
1
x+rJ
(16)
.
I!enee,
n
+1)
F(
'-
-
=
J.
L
+1'
ni
or as
Y
is a
dtnînny
variable we have formula 10,
n
=F(n +
1)
ni
As the trigainina, tetraga!runa, penthgamrna, et cetera functions
r(x) we have
are defined as successive derivatives of log
from
(16):
00
F(x)=
's
.T
1-= 1 (x
+Y)
00
f(x)=-2
-r= i (x
1
f(x)=6
r:;1 (x
- - e
Proceeding as with
+T)
+Y)4
------
f(x) we obtain formulas U,
ñ'om the assnnptotic expansion
the aSsyìnptotic expansion for
foi'
r(x +
12, and 13.
1), (34), we obtain
12
f(x)
=nx(x
+1) +
1(1
1
1
+
(x
i
+ 210
(x
5)- s..,
(17)
+ 1)
and for f(x),Rx), et cetera, by the derivative relation between
these finctions.
A table has been prepared which
f(x) for real
ives
Rx),r(x), f(x),
<x
x in increments of 0.01 for O
of 0.1 for 10 <X
=
<60
=
(2,
and
< i and increments
This table used with the
p.43-59).
following reduction forrnuias, (2, p. XIX-XXII), yields considerable
accuracy:
n-1
lC(nX)
=iog n +
f(.c) =((x -
f(x -/).
1) +
ir
ctn
inc.
n-1
/(rD:) = -.
n
: rc
-i).
(18)
f(x -
f(.x)
ein
1).
1cc
n-1
-ç-
r
ç-
L
(rix)
9(flX)
i
=7
n-i.
:
f(x -/).
if
Tables are now being computed for
ana
and
digaa
functions
for inaginary arguments (Ref. 7).
Formula 14 is proved by reduction of the
suimriand to
type form:
13
i1
;i.
x2+a22ai
i
i
L(x
-i
(x+ai -l)+lJ'
-ai -l)+].
and application of formula 10.
Note that formulas 10 to 13 are the one exception to exact
formulas in this chapter.
However, they are exact in the sense that
the sums are in exact terms of a defined tabulated function.
Lote that f(x) plays a role in finite calculus analogous to
that of 9n(x + 1) in infinitesimal calculus.
Exponential and logarithmic functions.
Formula 15 is proved by inversion of
(a
- 1)aX and
This division is by zero or inipossible if a = 1.
division by a - 1.
n r(x) = Rn x.
Formula 16 is proved by inversion of
1'rjgonontrjc and hyperbolic Lunc tions.
Formulas 17 to 20 are proved by inversion of the following
A-
relations, adjustment of the argument,
constant coefficient
arid
division by the
e
LXcos ax = - 2 sin(ax +
gin ax = 2 cos(nx +
)
a
sin
sin
)
a
(19)
Acosh
ax = 2 sinh(ax +
¿sinh ax =
.
2 cosh(ax +
)
sinìi
)
sinh
a
..
a
Combinations
of elementary functions.
Fornulas
2].
previous results.
to 32 are derived by summation by parts and
Evidently recursion formulas 30
arid
32
must be
used as a pair.
Formulas 33, 34,
arid
35 are proved by the substitutions:
[sin(a + b)x + sin(a - b)xJ
sin ax coe bx
,
[cos(a + b)x - cos(a - b) X]
sin ax sin bx =
,
(20)
and,
cos ax cos bx =
cos(a + b)x + cos(a - b)x
and application of formulas 17 and
Fórmulas 36, 37, and 3
and 35 respectively.
are
,
l.
special cases of formulas
33, 34,
They are of common application in practical
Fourier analysis (4, p.123-129).
Formulas 39 and 40
aro proved by the substitutions,
cos 2 ax,
(21)
and,
cos2ax =
+
cos 2 ax,
and application of formulas 17 and 16.
Note
that
analogous formulas for hiperbolic functions may be
easily derived by substitution through the relations,
sin(iz) = i
sirth z
(22)
and,
cos(iz) = cosh z.
IV.
STJMM.TION 13! METHOPS OF APPROXIMkTION
- relation or
Inversion of a
is possible
relatively
in
any
few cases as
exact evaluation of a
for
iO
analogous
of a derivative relation or any exact evaluation of an
calculus.
infinitesimal
Hence as with
infinitesLaal
evaluation must often be through series exansions.
those
assymptotic series so their
series are
sum
is
sum
inrsion
integral in
calculus
In summation
apnroximated
their convergent part, as in the case of the expansion of gimia
and related functions. Of course convergence and the remainder tern
by
must
alvys
he
investigated.
Newton's formula.
By
Newton's formula
and cnn terni
by
we may
tern. It states
expand u
in a series of factorials
that:
k
(k)
L2U° k.t
kD
For the proof
we
assn
u may be expanded into a series of
ol
uX = a + a x + a2x(2)+
and
(23)
.
evaluate the coefficients by setting x
factorials,
a3x ...
=
O
in successive
differences of this relation.
Extension of partial summation.
If wo let r41.
in the fourth form of formula
5 w
have formula
16
Formulas relating a
to an integral,
sum
over
The sum and integral of u
related as may he seen by geometry.
the curves
Hence
Y
%,
a
the
sum
correction terms.
Thc integral is th
area under
is an approximation to this
may be expressed as the
a siin
interval are closely
the same
area.
corresponding integral and
If these terms are in the form of differences,
the formula is Laplace's type, and if in the form of derivatives,
Euler's type
(6,
p.104).
Formula 42 is of Euler's type and is called the
formula.
It
in
derived by expanding the operational equivalent of
(Q,d/dX...l)_]\,
''
B+i which
The
in powers of d/dx.
appear in the coefficients
rnnubers and are
Euler-MacLaurin
nwnbers of
The
the form
are called Bernoulli' s
of froquent occurrence in approximate sum formulas.
recursion relation:
n-1
E
(-1)(')B
i
relation is consistent with the definition,
list
of Bernoulli's
(24)
Bernoullian numbers
may be used to evaluate the
brief
= O
B2
numbers is included
(4,
p.233).
=
for n
O
later.
This
This
1.
A
list is
sufficient for most practical purposes because the magnitude of the
numbers increases so rapidly that any assymptotic series whose
coeffIcients contain them has a small convergent part.
Formula 43 is of Laplace's type and is called Gregory'a formula.
It is derived
by substitution for the derivatives in formula 42 of
their difference equivalents
(,
p.62).
17
Sum from every m'th term.
We
have a Woolhouse or Lubback type formula according as
the correction terma in such a formula are in terms of derivatives
or differences respectively.
Formula 44, a boThouse formula, is
derived from formula 42 by eliminating the integral between formula
42 and a formula derived from
m parts.
42 by subdividing each interval
into
Formula 45, a Lubback formula, is derived by replacing
the differences in 44 by their differential equivalents.
Formulas
to
x,
46, 47, ax1 48
1/x, and l/x2
are derived by applying
respectively.
Formula 49 is discussed in Chapter VI.
formula 42
TALE
V.
i.
2.
OF
S1Th
c=cx+c.
2111
e
f(x).
x) =
f[r(x)
]
±t:(x)
1n+1
n
u11
4.
f(x)
=
ini
'ni-1
XIIm
X
=
X
z
g(x) ±
ni-1
¿
-
I
nl
±
mi-1
n
-
ni
mi-2
¡ni-1
y
xl
+2ux-2
u
x-1 ¿
i
In+3
Z. -' ir
i
J
ni-r
.,
= u
X
E
V
X
-u
E
rn-fr
2\
X
z
+
(1)rL
:
x
n
In
v
-T
n
u
=
nl
ni-1
3J.
xi-2
¡
n
u
In
x
x+r.
ni
5.
x-r
X
V.
(ni-1)
6.
E(x
+
b)(nl)
a(: +1)
3
i-
nl
?ero or any integer
-
1.
19
i
4(x)
2:
..
Ai Z
"x1c+r
1
ax + b,
integra], function of degree
of
(x), 4'(x)
Ç)
2:
=
9
Gm
a,
Rn
Au+i
+
- i, A's from expaneion
...
+
z integers such that O <a
x
L=iJ
1 =
10.
+ C;
A0ux+
ni
;
a rational
cj)(x)
<.
z.
on page 25.
Values of
+ i)- f(m); f(x) =
DQn ["(z
+ i).
n
f'(n + 1)+f(m); j'(x) = D f(x).
i].
ni
(x+i)2
n
12.
4:
ni
n
1:3.
x:
ni
=
(z
r(n +
i) +
f(m); f(x) =
+i)
tn4a
i
14.
Im
X
15.
ZaX
a
a-i
C; a
+
i.
ln+1
__
-2f(x-1+ai)I
Im
20
16.
17.
1L
=nr(x)
2n(x)
sin(ax
+ b)
Zcos(ax + b)
+ C.
Cos(ax +
-2
-
b
a/2)
+
sin a/2
-
sin(ax + b
a/2)
+
c.
2 sin a/2
19.
sinh(ax + b)
cosh(ax + b
sinh
2
20.
Zcosh(ax + b)
21.
xaX
22.
X
(ni)
a/2
sinh(ax + b 2 sinh a/2
a
x
a
a
-1 -
-
(a
aX
=x (ni) a-i
+
1)2
X
(ni)
a/2)
+
C.
ma
X cos(ax
-
8inax=
sin ax
aJ2)
2 sin a/2
24.
a/2) +
X
=
sin
23.
-
x(m)cos(ax
- a/2)
____________
sin a/2
2
m(m
-
+
+ 4 sin2a/2
1)
nix
+
(m-1)
4 sin2a/2
X(m_2)sin(ax +
4 sin2a/2
25.
x cos ax =
X
sin(ax - a12)
2 sin a/2
cos ax
+
4
sin2a/2
sinax
+
a).
21
Xs
(in)
26.
Zx'cos
- a/2)
1r«
2 sin a/2
ni(m - 1)
in
+
4 sin2a/2
Zx(m_2)cos(
+ a).
2
4 sin a/2
In formulas 27-32:
acosb-1
Ka2+1_2acosb
'
a cos b
a2+1_2a cosb
27.
2.
29.
aXsin bx = lÇ1aXsln bx - K2aXcos bx + C.
ZaCcos
bx = K2aXsjn bx + IçÍaxcos bx + C.
xasin bx =
(K1- K2)xaX(sin bx
+ cos bx)+(I- 4)a''sin(bx-th)
+ 2 K1yXtbcos(hc + b) +
30.
x
(m)x.
aslnbx=K1x
-in K1
31.
(m)x.
asinbx-Kx (m)x
acosbx
Zz(m_asin(bx
xa%os bx =
(K1+
x+l
a
C.
jç)X(5j
.
4b)+ mK
X
(m-l) x+1
a
cos(bx + b).
bx + C05 bx)+(K+
cos(bx + b) + C.
K)
22
32.
ZxaXco
naK2
bx = Kx(m)axsin bx + jç1x(m)axcos bx
Zx(m1)a1sin(bx
+
cos[(a+ID)x
33.
sin
ax
x1a1cos(bx
b)- mX
cos bx
a+bl
34.
Zain
ax
sin
bx =
-
[(a..)x
48m
sin [(a-ö)x
cos ax cos hx =
:35.
36.
k
L
k
sin ax1cos
37.
z
:3e.
x:
39.
: sin2ax
40.
os
2
:
bx=
in axsin bxi
k
io
sin
axcos
= 4c
ax =
1
-]
]
a) +
a
a-1
+ C.
4am
a-k1
.'
- -j--J
sin [(a-bix
A
= O.
sin(2ax-G)
+
-
+C.
4
bx1 = O.
x +
a.b)x
sin
+-
O; a, h integers
- sin(2ax
a
in
4
+ C.
a-b
4sin--
2
_
b).
eo8(a-.b)x
-j--J2
sin
+
C.
<
k,
x1=
-
1ì
+
2d. .
-i-
23
2(2
00
v=
41.
3.
.+(.)r+1ôr-1
+
.'.
d3u
'
ux =
-
5u&
C +
iu +
2 x
+
dx
L
I! dx3
1du
=
C
+
-
ValueB
2n+1
n
dx
12
1x
d5u
+ 30,240
720
+)rn
m
udx
n/rn
u= m
+
I
m-1
2
3f2+
ml
[un+
UJ
n
du
2
-
i
d\
n/rn
45.
in
2
24mL
(in2- l)(
-
rn-i
n-2
N
+A2
-
(°
in
+
I
r/203
n
-
[íxu+ ó2u]
+
i
im
+
n
5
on page 25.
(n+].
=
44.
1d3u
dx3 lo
2
m-1
n-1
Au
(m2-1)(19m2-i)
720m3
o
m2_I[A4
+4u}
-
3
In-3
o
46.
C
47.
ZC
4i.
+
-
+
(2n-1)x'1
00
n(n - 1)(n - 2)
+---2x
-
r=).
[ rn.
(1z)
72Ox2'
1
(1)1()(ra+1)fl]
1nk_ki
-
s-logx
2-1-=c
-¿rl
LJ.
=
i
?
25
G
Stirling's Nwnbers of the Second Jind,
i
i
i
2
:i
2
3
4
5
6
7
1
1
3
i
3
1
4
i
7
6
1
5
I
15
25
10
1
6
1
31
90
65
15
7
i
63
301
350
140
21
1
127
966
1701
1050
266
2
Bernoulli's Nuribers
B0=
1
-11
691
2,730
i
36i7
1
133
=55
x
15
1
17
1
510
43.867
798
174.611
330
137=55
B
21=
854.513
138
26
STJM&TION OF A SPECIAL TYPE POWER SERIES
VI.
The sirnation o1 a power series in which the coefficient of
the k*th term i
a rational integral function of k is 8tudied on
the following pages.
By sunniation
iiean
here, the replacement
of the power series or infn1te sum by a finite expression, identical
for values of the variable in the intercral of convergence of the
por
series.
In other
rds the problem of approximate evaluation
of the infinite series is reduced to exact evaluation by
substitution in an identical finite form.
We proceed to devalop thin finite expression.
Also we develop
three recursion relationships 'which are of more practical use and
which iield identical resuJts for this type of sination,
Finaily
e present a triangle, the sutnnation triangle, which, with a simple
forirmla,
nt
oily provides the most rapid method for such summation,
but is also easy to remember.
The general form of such a series being written:
(C0k+
s x)
there the
Cs
C1l+
+ c)xk,
(p5)
are constants and n Is a non-negativa integer, we
have immediately the form:
S (x )
=
C
Ol
:
kxk+
c
+
:
f
'.
¿
1cl
Thus the stated problem of summation may be reduced to the problem
of evaluat4ng sums of the form:
=
ithere
E
k1xk,
(26)
In other words rather than
n is a non-negative integer.
studying the original power series (25), we str1y the power serios
(26).
K(x),
The interval of convergence of
(-1,
+ 1), may be
readily determined by tho ratio test.
Then n = O, we have:
K(x)
which is a geometric series.
1Ç0(x)
We see
Çx)
Now
Hence,
=
/x/<l.
-
(27)
geometric series.
is a particular
oceeding from (27) we have:
lk=
I%(x) =
c1
(x)
=
°
k=l
(l-c)
(28)
00
JL(x)
¿
=
:
2= :K4
(x)
= x(1 + x)
kl
(l-x)3'
and in general z
00
nlc
xP(x)
.1
kx=xç_1(x)=
K(x)=.
n
k=l
may be seen to be a polynomial of
deee
n
I by luduction.
')n
Note that P2(x) is a polpiomial of degree one and that if
is a polrnoinial of degree n
of degree n.
1, then
?1(x)
P(x)
will be a poljnomial
20
Since K(x) satisfies the recursion relation,
Ic1(x) = x K
(29)
(z),
and since,
X P
(x)
K(x)
(1 ..x)
(30)
ii+Ï'
have,
x?,(x)
dfxPn(x)_1
X)saJ
L __
(1
'
vhence we obtain, by perforning the differentiation and removing
X
the factor,
(1 ..x)
P
(x)
s
(nx
n-s-1
+l)Pn (x) +x(1
P,
The successive polynomials,
easily by repeated use of (31), starting
P1(x)
=
x)p1 (x).
n
P3S ...
froia th3
,
(31)
can be built up
known fact tbàt,
Thus wo haves
L1
P1(x)
= i
P2
=i +X
P3(x)s1+4x+x2
P4(x) s
P5(x)
s
-s-
(32)
lix + 11x2+
26
+
¿
26x+ x4
It may be noted that the coefficiente in these polynomials are
smmetrica1, positIve, integral, and that the first
arid
last
coefficiente are unity.
If we assune that:
:Pn(x)
fr(fl)Xr_l,
=:;l
(33)
29
and substitute in (31) we obtain the recm'sion relation for the
coefficients fr(n) as follows:
=
(n - r
Since Ç(ii)
-O,
ie see that
f(n) = i
+
1)1'
r-1
(n
1)+ r
r'
we see that f1(n) = i for all n.
for all n.
- 1).
Since
(34)
f1(n) =
O,
Hence we see that the first and
last coefficients are always unity.
The other
the coefficients follow easily fron (34).
iroperties noted for
Their stretry is shown
later, (47).
Using (34) w
build up a triangular array o± coefficients
for the polynomials P(x), shown on page 30.
Looking at the geometrical neaning of (34) with respect to the
triangle we find a de'vice whereby the law of formation may be easily
applied and remembered.
triangle.
This law is similar to that of Pascal's
Rather than adding t
elements to obtain the one below
we add multiples of the elements.
by the number of steps we must take
to reach the elements.
The
These multiples are determined
frein the sides
of the triangle
ìethod is evident from (34).
uewdoOAej
;o
UO?tLUfl UÇ.ZJ1
eI{'4
.zoJ
eut ewçi c
E)q'4
t
-t
-i:
t
1
i
t
ri:
99
9
t
t-t
9
t
t
t
t
ot
t
t
9t
t6tt
6t9't
O9't
t6tt
ot
t
6t9'I
O6t'9t
t
O9't
t
Cr'
o
31
The next step is to find a finite expression for
'iVe
have by repeated application of (34):
(n
=
(n
=
(n
-r
+
1)Ç1(n
1)
r +
-r
+
1)f1(n
1)
i»
r(n - r)fr 1(n - 2)+
1)-s.
r(ri
-r -1)f1(n
+ r2(n
n
+ r
-r)fri(n
3)
f(n
- i)
= 1 when n
+
- i
i<n
= r or when
i
-1.
= n -
Hence,
=
r(n
-r -i
-
+ 2r-.i(
i» r"'Ç,(r),
(r - 1),
or since f (r) = f
r-1
r
n-r+l
f (n)
r
=
r'
(n
-r -i
+
2)Í'r
Now we substitute r = 2 in (35) and note
i
<n
(n - i)
(n - i)
(35)
= i for
to obtain,
n-].
2'(n
f2(n) =
- i).
i=].
Summing by parts, formula 4, form 3, in the table, we obtain
f2(n), f3(n), f4(n), and f5(n) successively, each time depending
on the previous result and noting f1(n)
- 2)
- 2)
+r'(n_r.4+2)Ç_1(n si)+r1fr(n -i);
Now since f(r) = I,
r2Ç(n
1,
to obtain:
r.
32
in
=
2fl(
3fl_(
f3(n)
4fl(fl
5-r
1'5(n)
+ 1)
+ 1)n
+ 1) 2"+
3n
+ 1)
(36)
2Z
1)n 2n
(n
(n
+ l)n(n - li
3Z
4n
+ 1)
(n
+l)n 3n
(n + l)n(ri - i)
2n
31
(n
+
+ i)n(n
L)(n
-2)
.
II
'4.
is evidently,
From this an intelligent conjecture for
r
F (n) =
r
(l)fh(fl+l)(r -ni + i)'.
inductive proof that
The fol1owin
()
ni-1
m.
Fr(fl)
consists in
=
'r
showing,
F(n) =
+
(n - r
l)F1(n
- 1)+ r Fr(fl -
2
r <
n
(3g)
and,
F1(n) = F(ri) = 1
in this order.
Hence
first prove:
ïe
(1)fli+l(fl+l)(
n
(
)(r-m)
n-1
1)fl_
-
-r
(n
r
+r
(-1)
n
rii+i
(
+ 1)
1)(r
-in+
(1)m+i.
1)1
m=1
We proceed to reduce the express4.
un the right to that on the left.
Changing the sunmiand of the first terni on the right sud adjusting
33
through the limits of summation,
r
(n
+
have the two
obtain:
(_l)mÇhl2)(r
i)
+
We
we
r
rZ
ml
+
(4)m+1(
1)fl1
)(r -in
relations:
_m-1
-n +i
n
I
n+l.,
Ç_i"
(39)
-n + 2
- n +i
n
n
'm-l'
,n
which follow imnecliatelç from the expression of the binomial
coefficients in their factorial form. Substitution from these
equations in the above expression ylelds:
(n
-r + i)
r
(_1)rn
+
m
+
If the
r
rL
ml
(_1)r
exponents of -1 and (r
_ (n
-r + 1) r
+
-i n+l
ni
r
-
n
nl
(_1)fl
i to r
,n+l
in-1
are adjusted, this becomes:
,n+l,
i)r
l)(r-'n+l)
n-in+2
-
in
sum may
in
=
+
ni-1
1)fl
just as well
be
(n+l)(r-m+i) ( in-i )(r-m+
as the term corresponding to
make this change, includo
)(r-rn+i) n-i
in-1
(l) iii+i
limits of siination for the first
from
ir w
+ 2
-n + i)
rn-_
The
in
n+l
O
is zero.
for
Hence
the constant factors inside the
swrxrnation, and add sums, we have:
s/t
r
çUpon
I (n-r+l)(m-1
(_m1
r(n.n+2)
(n+l)(r-n+l1
+
(n+1)(rn+l)j
(:)(r
-
in
algebraic reduction this becomes:
mL
and the
first
To
(
-))fl1+l
equation of (38)
continue,
we
is
,n+l
1)fl,
m-1
proved.
evaluate F1(n) by substitutIon in (37) as
follows:
F1(n)
Finally we show
(_l)2(1)
Fa(n) = 1.
To do
(40)
this
it will
be convenient
to consider an array which is our triangle augmented by the elements
F(n - 1),
n
.
F(n
-
1) must be defined
Hence
to
satisfy
2, thich
by (37) as this
defition
-
=E
1)
(_1)m+l
=
m +
1)nl
1,
=
(E
-
which becomes by the binomial formula,
(-l)'
If
we
apply the
causes it
Thus,
(33).
F(n
Now since
satisfy the recursion relation of our triangle.
E
operator
n+l
zm=l
we
(_1)in
have,
(1)(x
+ n
-n
+
1)fll
(41)
35
Hence we have,
n+].
Afl n-1
1)XTI+1
I
xD
or as the term Lor
ra
'x'I
(
m1
n
,
n-1
)I%n-m+1j
,
m-1
= n + i equals zero,
(1)m+1
IxJ
(
-
=
-m
(rn1)(n
+
(42)
m=l
p.l9-20)
(1,
If we compare (41) with (42) we have:
F(n_1)=xhf
n
lx
Iow
an-1
=
(n - 1)
means
n
n-1
=0.
¿
X
F(
n - i) = 0.
Ience we have:
Now from the proved recursion relationship of (38),
F (n)
n
= F
n-i
(n - i)
then r
+ n F (n - 1),
n
or as F (n - 1) = 0,
n
F (n)
n
=Fn-J. (n
-1).
Since n i3 arbitrary,
F (n) = F
n
n
1(n - 1) =
F(n
- 2) = ... = F1(1).
Now since F1(1) = 1,
F(n) =
i.
(43)
36
This completes the proof that,
f(ri)
=F(n),
or that,
()Iìi+l
Now with f(n) we have from (30)
arid
(')(r -m
(33):
(-i)()(r
i
(44)
m + l)vEJxr_1, (45)
[
or in a slightly different form,
r1,
,
knXk
k=].
i
m+in+i.
n
Çn_ii(1)]
r
x
I
(46)
(1 -
The application of ali this information about (26) to (25)
is Lmnediate.
37
A swmary of the results of th1
of their application is now in order.
investigation and a discussion
The methods and relationships
mentioned at the first of this chapter have now been developed.
have the finite expression of the swn (25) by applying (45) or (46).
Wo have the recursion relationships (29), (31), and (34).
The
best method of evaluation of the swn is by use of the stwnation
triangle with (30) and (33).
In the suirnnation of (25) the value of
methods depends on
The magnitude of n, the proxidty of ¡xi to unity,
the following:
deee
and the required
of accuracy of the evaluation.
considerations must be given each series.
siaU but ¡xi
oi.r
Indidual
For example, if n is
is near unity our nethods are of value, but if n is
large and ¡xi is snall the convergence of the series is not
seriously slow and our results are of less value practically speaking.
The evaluation of (25) for sorne particular value of x may he
accomplished in various ways such as direct substitution in a finite
niiher of terms or by expression of the polmornial coefficient of
xk in factorials followed by sunnation by parts.
JJovevr, the
methods presented here are of gieatest value for this evaluation.
For clarification we give typical examples where these
methods are ìarticu1arly suited.
First, let us evaluate the infinite power series,
A
J.
+ x +
2+ 27x3+ 64x4+
for x = 0.90000 to five decimal accuracy.
...
+ k3xk + a..,
The exact value may be
found by Í'orrnulas(30)and(33)and the summation triangle and substitution
38
îor x.
Thus we have,
s=i+E
k3xk
(-l)4x
=
=
2.2
=
+
Again
using
+
c
+ x2)
43,69l.
Second, evaluate the convergent irif:Lnite
S
(1 +
-
(z
3.22
332
2.3
2
2k + 3k2+
series,
k4
+
34
+
...
forrnulas(30)and(33)and the suimiation triangle, we have:
2k
=
+3k2+
i/
k()k3
=2
k=l
+
k=1
L
= 172.
Finally,
infinite series
finite expression for the sum
k'th teri!l1 is the product of the k'th
let us find a
whose
of an arithmetic progression, b, b + d, b + 2d,
and the k'th term of a geometric
proession,
The series is convergent for r <1.
a,
...,
ar,
b + (k
2,
of an
term
-
1)d,
ar.
:39
ab
(b
+ kd)ark
1k
rk+ ad
= ab + ab
-a b-br+dr
(1
A study of the
siation
interesting
Sorne
immediately
frani its
First, the
- r)'
triangLe.
'operties of the summation triangle follow
law of formation.
elents
of each ro
are syrìetrica]. about the
element or elements at the middle of the row.
=
i
n-r+i'
r
Ç.
The proof is inductive with respect to n.
n = 1.
<n.
(47)
It is evidently true for
If it is true for n1 we have the two relations:
(n
-r
+ 2)2
1(n)
=
(n
-r
+
2)f(n),
2
i
n.
= r
r
I_f
That is,
r
r
we add equals to equals in the above relationships,
(n
-r
+
2)21(n) +
r f (n)
r
r f
2
<
=
r
n-'r+l
(n)+(n
-r
=
n-r+2
<
= n.
1), 2
r
e have:
+ 2)2
Now by (34) we have:
1)
+ 1.
n.
n-'r+2
(n),
40
Now
since f1(n
+ 1) =
f1(n
r <n + 1. Thus the
(47) is complete.
J.
Socond,
this relation holds for
+ 1) = 1,
for n
symmetry holds
+
i
and the proof of
for each line the elements increase from the ends to
the middle, that
is,
f(n) <
1'
noven
It is sufficient
first inequality as the second
first by the smetry just proved.
proof is inductive with respect to n. Evidently the
inequality is true for n
=
holds for n odd and prove
it
will
Now we
3.
n
fr(fl)<fr+i(fl) for r
inequalities:
Hence
we
multiply the inequalities by
add, and use (34)
we
r
for
frorii the
srnuetry.
=
this inequality
and n odd ç4ves us the
<fr+i')'
r,
r
n
+
1, and 1 respectively,
obtain:
n even we use the
-i and
i
er-i
1), r
r+i
Now
;
r_i<r'
fr<1r+i(
assume
holds for n + 1. Repeating for n even
completos the roof.
If
(48)
to prove the
inequality follows from the
The
j,n_.l nodd
=
+(r)> f_(n)
relations
for r
Proceeding as above we
<fr+i
+
r
for
fr')<fr+i
This
=
i
n +
latter
equation is
obtain:
(n +
l
-
i
41
Thus
the induction is comleted and (48) is proved.
An Abel sum.
if
a
series diverges,
it may he
first
combination than the
(9,
sums have a
These sums are
p.261-265).
definition and not by analogy with
cbviously these
other
some
terms as an approximation to the
i
"sum" of the divergent series
up by
possible to use
different
of convergent series.
sums
meaning than
in
the case of
convergent sanes, but they aro often of practical value.
true especiafly when this sum is regular,
to obtain the
Now we
p
fl
sum a
from the
that is, the
definition of
-i in the
Abe1
This is
methods used
convergent series to the ordinary sum.
have an example of Abels sum for the
obtained by setting x =
set
oscillating series
expression for }Ç1(x).
We
have
sum,
A =
and (46),
00
:
(_l)kkfl
=
n
lini
x-p-1
(1
-x)1
r
;;i
I
()(r
(_l)m+1()(r
=
':;i
by the method of Abel.
=i
-
(_1)m+1.
;:i
-
mn
ni
+ i)7J
r
+ l)nJ (_1),
(49)
42
K(x) for negative integral
To
new
some
gain perspective on our previous results
results
n =
1,
n and O.
we
and
to obtain
some
will develop a finite expression for K(x) for
develop a method whereby K(x) may be approximated for
particular
-2.
'value of x where n
relation (29), evidently true for
initial condition,
K0(x)
e
will use recursion
any complex number n,
in the form:
klxk,
kflxk = (xD)
II
with the
arid
(50)
, for this extension.
=
Hence we have,
kxk
= (xD)
-1
k=l
k
kl
X
= (xP)
=
(dx
)(i -x)
= -.2n(l -x) + C.
Since
klxk
O
for
O,
X
l
C
klXk
= O.
This gives,
-.Qn(l .ix),
(51)
The known expression for the sun of this common series.
For n
in the
-
-2 we obtain in the same way a
form of an
integral,
an expected
non-finite ex'ession
result for the
sum
of such
43
a
series.
It is convenient to consider the
form with limits
O
and
x.
The
constant
E,-2k
c
X =-\
/o
k1
in its definite
drops out and we obtain:
Pn(1-x)
X
x2n
=
(52)
f
X
I
10
Repeating
intea1
this process for successive negative inteers,
obtain a multiple integral of multiplicity (n - 1) for K(x).
ve
:re
have:
E.
kxk=
kl
(x
(X
/0
I0
From a practical
10
11n
x
1x
()fl1
(3)
standpoint approximate values of these
integration of (53).
series may be found by numerical
of (53) decreases as ¡n! increases for
infinite
The utility
two reasons, the integration
bocomes more tedius and evaluation by other means becoies easier
due to the increasing convergency of the series.
series converge
for x
= -1 when n
- 1.
Note that these
4h
BIBLIOGRAPHY
VII.
1.
2.
Bode, George. A treat1e on the calculus of finite differences.
3d ed. New York, G. E. Stechart arid Co., 172. 336p.
British association
tables,
72p.
for the advancenent of science.
vo].. 1.
London, The Cambridge
Mathematical
University press.
3. Fort, Tomlinson. Finite differcnce ai difference equations
in the rea]. domain. London, Cc.ford University press, l94.
251p.
4.
Jordan,
5.
MIme-Thomson,
Charles. Calculus of finite differences.
York, Chelsea publishin; co., 1947.
652p.
L. ì.
The calculus of
London, Maca11an and
6.
Steffenson,
J. F. Interpolation.
.lkine co., 1927.
7.
co., ltd.,
finite
1933.
New
2d ed.
differences.
558p.
Baltimore, The
Williams and
248p.
Mathematical tables project.
and digaurna function for complex arguients.
Now in process of computation.
U.S. Bureau of Standards.
The
ga!nna
.
Whittaker, E. T. and Robinson, G.
Calculus of observations.
1944. 397p.
4th ed. London, Blackie and eon,
9.
Widder, David. Advanced calculus.
1947. 432p.
ltd.,
New York, Frentice-HaU
inc.,