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The Sigma Summation Notation
#18 of Gottschalk's Gestalts
A Series Illustrating Innovative Formsof the Organization & Expositionof Mathematicsby Walter Gottschalk
Infinite Vistas PressPVD RI2001
GG18-1 (28)
„ 2001 Walter Gottschalk
500 Angell St #414Providence RI 02906permission is granted without chargeto reproduce & distribute this item at costfor educational purposes; attribution requested;no warranty of infallibility is posited
GG18-2
N
thus
.
•
files
• the file of length n (n )
= the n - file
= n
= n file
= the set of the first n positive integers
= {1, 2, , n}
= [1, n]
• the file of length zero
= the 0 - file
= 0
= zero file
= the empty set
=
1 = {1}
2 = {1, 2}
3 = {1, 2, 3}
etc GG18 - 3
dn
rd
df
dn
rd
df
Œ
◊ ◊ ◊
∆
∆
P
P
0
N
a
of elements a
a
family a i n
a
a
i
.
, ,
( , , )
( )
( , , )
(
the basic capital sigma summation notation
• the sum
a + a
= a plus a plus plus a
a , a (n ) of an additive semigroup
= the (termwise) sum of the ordered n - tuple a , a
= the (termwise) sum of the n
= a , a
= summation / sum of
2 n
rd 1 2 n
2 n
2 n
dn 2 n
rd
1
1
1
1
1
+ + ◊ ◊ ◊◊ ◊ ◊◊ ◊ ◊ Œ
◊ ◊ ◊- Œ
◊ ◊ ◊Â
P
,, , )
( )
( )
var
a , a
=
= summation / sum of
=
= summation / sum for i in n of a
=
= summation / sum from i equals 1 to n of a
n or at least i is an integer - valued variable
that contains n in its range
2 n
dn
rd
dn
rd i
dn
rd i
◊ ◊ ◊
Œ
Œ
Œ
Â
Â
Â
Œ
=
a i n
a i n
a
a
wh i
i
i
ii n
ii
n
1
GG18-4
N in e sigma summation notation
a
i
i is called the summation index
C
corresponds phonetically
ii
n
.
' ,
)
.
&
th
is a bound = dummy = umbral variable,
' (variable)
the set of values of i used in the summation (viz n here)
is called ' the summation range' ,
a (i n is called ' the ith summand' or ' the ith addend' ;
the summation index variable is often taken to be i
because i is the initial letter of ' index' and of ' integer' ;
summation index variables
are frequently taken from the alphabetic run h, i, j, k
and other good candidates are m, n, r
in 1755 Euler introduced capital sigma
to denote continued sums;
the indicial notation was added later by others;
the Greek letter sigma
i
=Â
Œ
1
S
Ssinin ansliteration to
the Latin English letter ess
tr
Ss
wi the initial letter of summa (Latin) = sum GG18 - 5
-
N
a a
wh
m
i
a a
are elements of an additive semigroup
m n
m n
.
var [
, , ,
extended capital sigma summation notation
• a
= summation / sum from i equals m to n of a
= a
, n st m n
&
m, n]
or
i is a variable
whose values are integers
and
whose range includes [m, n]
&
a
ii=m
n
rd i
df m
m
Â
+ + ◊ ◊ ◊ +
Œ £
Œ
◊ ◊ ◊
+
+
1
1
Z
Z
Z
GG18-6
• )
)
)
)
the (termwise) sum of (a
= (a
= summation / sum of (a
=
= summation / sum for i in I of a
=
(a is a nonempty finite family
in an additive commutative semigroup
&
n = crd I
&
: n bijection
note: if I is also a totally ordered set,
then is uniquely definable as an order - isomorphism
&
the commutativity of the semigroup may be dropped
i
dn i
rd i
dn
rd i
df j
i
i I
i I
i I
a
a
wh
i I
I
ii I
j n
Œ
Œ
Œ
Œ
Æ Œ
Â
Â
Â
Œ
Œj
j
j
GG18-7
• the (elementwise) sum of A
= A
= summation / sum of A
= (a a A) = a
semigroup
dn
rd
dfa A
Â
 ŒŒ
wh
A is a nonempty finite subset
of an additive commutative
GG18-8
N
a
a
a
are elements
m
ii
m
.
, ,
the sigma summation notation for series
• the sum of the right series
a = a
= a
= summation / sum from i equals m to infinity of a
= lim ( n var [m, ) ) iie
wh
m
&
a
of an additive topological semigroup
m ii=m
dn ii=m
rd i
dfn
=m
n
m
+ + ◊ ◊ ◊
Œ Æ
Œ
◊ ◊ ◊
+
•
Æ•
+
Â
Â
Â
1
1
Z
Z
GG18-9
•
var
,
the sum of the left series
= a
= a
= summation / sum from i equals n to minus infinity of a
= lim a ( m ( , n] ) iie
wh
n
&
,
elements of an additive topological semigroup
i
i=n
dn i
=n
rd i
dfm
ii=m
n
◊ ◊ ◊+ +
Œ ¨
Œ
◊ ◊ ◊
-
-•
Æ-•
-
Â
Â
Â
a a
a a
are
n n
i
n n
1
1
Z
Z
GG18-10
•
var
, ,
the sum of the biseries
+ a = a
= a
= summation / sum from i equals minus infinity
to i equals (plus) infinity
= lim a ( m, n & m n ) iie
wh
, a ,
elements of an additive topological semigroup
ii
dn ii=
=
rd
dfnm
i=m
n
◊ ◊ ◊ + + + ◊ ◊ ◊
Œ £
◊ ◊ ◊ ◊ ◊ ◊
-Œ
-•
•
Æ•Æ-•
-
Â
Â
Â
1 0 1
1 0 1
a a
a a
are
i
i
Z
Z
GG18-11
N
summation sum from
C n n
.
/
( )
the sigma summation notation
with two or more summation indexes = indices;
here are three examples
• a
= summation / sum from i, j equal 1 to n of a
= a sum of n summands from a to a
• a
= summation / sum from i, j, k equal 1 to n of a
= a sum of n summands from a to a
• a
= i, j equal 1 with i less than j
to n of a
= a sum of =1
2 summands from a to a
iji, j=1
n
rd ij
211 nn
ijki, j,k=1
n
rd ijk
3111 nnn
iji, j=1i< j
n
rd
ij
n 12 n
Â
Â
Â
-2 1 --1,n
• it is understood that n is a positive integer & that the a' s
are from an additive commutative semigroup GG18 -12
D laws for elements a (i var )
of any additive semigroup
• special range laws
a = a = = 0 iie
a = a
a = a + a
a = a + a
i
i=1
0
ii
df
ii=1
1
1
ii=1
2
1 2
ii=1
3
1 2
Œ
∆
+
  Â
Â
Â
Â
Œ∆
P
i
a
etc
3
GG18-13
D law for an element a
of any additive group
• the right distributive law for multiples
= (n & for i nii=1
n
ii=1
n
ia a a ÂÊ
ËÁ
ˆ
¯˜ Œ Œ Œa a P Z )
GG18-14
D laws for elements a (i var )
of any additive commutative group (n )
the negation law
=
• the left distributive law for multiples
= (
i
i=1
n
ŒŒ
- -
Œ
= =
=
 Â
 Â
P
P
Z
•
)
a a
a a
ii
n
ii
n
ii
n
i
1 1
1
a a a
GG18-15
D laws for elements a var
of any commutative ring (n
square - of - sum law
= a
co square - of - sum law
= (2 a (a
i=1
n
ii=1
n
i=1
n
ii=1
n
ii, j=1
i
i ii ji j
n
j
i
i j
the first
a a a
the se nd
a n
( , )
)
•
•
)
.
ŒŒ
Ê
ËÁ
ˆ
¯˜ +
Ê
ËÁ
ˆ
¯˜ - +
  Â
 Â
=<
P
P
2
2
1
2
2
2
ii< j
n
i=1
n
ii=1
n
i, j=1
i< j
n
square - of - sum law
= a (a
Â
  Â
+
Ê
ËÁ
ˆ
¯˜ - -
a
the third
a n a
j
ii
j
)
•
)
2
2
2 2
GG18-16
D a b (i var )
of any additive commutative semigroup (n
• the additive law
(a = a b
i i
ii=1
ii=1
ii=1
law for elements
bn
i
n n
,
)
)
ŒŒ
+ +Â Â Â
P
P
GG18-17
D a b (i var )
of any additive commutative group (n
• the subtractive law
(a = a b
the linear law
( a = a b ( )
i i
ii=1
ii=1
ii=1
ii=1
ii=1
ii=1
laws for elements
b
b
n
i
n n
n
i
n n
,
)
)
•
) ,
ŒŒ
- -
+ + Œ
  Â
  Â
P
P
Za b a b a b
GG18-18
D the binomial theorem
in any commutative ring (n )
first form
(a + b) = n
i
• second form
(a + b) = n
i
third form
(a + b) = n!
i! j!
which has three indexes
in general
multinomial theorem which has say r 2 indexes
n
i=0
n
n
i=0
n
n
i, j=0i+ j=n
n
Œ
ÊËÁ
ˆ¯̃
ÊËÁ
ˆ¯̃
≥
Â
Â
Â
-
-
P
•
•
:
&
a b
a b
a b
note the third form suggests
the trinomial theorem
the
i n i
n i i
i j
GG18-19
D B
are definable by a generating function
x
e
x
e
x
n
whence
B B
B
x
x
n
. , , ,
!
, ,
,
the Bernoulli numbers B B
(x real var)
as follows:
= B (for x near 0)
B 1, B = = 1
6 = 0, B =
= 0, B = 1
42 B = 0, B =
0 2
nn=0
0 1 4
6 7 8
1
2 3
5
1
1
1
2
1
301
◊ ◊ ◊
-Œ
-
= - -
-
•
Â
3030,
,
B = 0,
B = 5
66
9
10 ◊ ◊ ◊
GG18-20
D the complex functions
exponential, sine, cosine
are definable as
everywhere - convergent power series
as follows:
exp z = = ( z )
sin z = z
= ( z )
cos z = (
n=0
2n+1
n=0
n
z
nz
z z
nz
z z z
n
n
n
! ! !
( )( )! ! ! !
)
•
•
Â
Â
+ + + + ◊ ◊ ◊ " Œ
-+
- + - + ◊ ◊ ◊ " Œ
-
12 3
12 1 3 5 7
1
2 3
3 5 7
C
C
==0
= ( z )•
 - + - + ◊ ◊ ◊ " Œz
n
z z zn2 2 4 6
21
2 4 6( )! ! ! !C
GG18-21
D an example;
the Laurent series of a complex function
that is analytic on the punctured plane
and
that has an essential singularity at the origin (z var )
exp z + exp 1
z
= e
= 1+z
1!
= z
1!
=
z
Œ
+
+ + + ◊ ◊ ◊+ + + + + ◊ ◊ ◊
◊ ◊ ◊+ + + + + + + + ◊ ◊ ◊
C
e
z z
z z z
z z z
z z
z
1
2 3
2 3
3 2
2 3
2 31
1
1
1
2
1
3
1
3
1
2
1
12
2 3
! ! ! ! !
! ! ! ! !
zz
n!
1
n!z
= z
n ! +
z
n!
= z
n ! + 2 +
z
n!
n
n=0n
=0
n=0 n
n=0
n= n
n=1
• •
-•
•
-•
- •
 Â
 Â
 Â
+n
n
n 1
GG18-22
N
a a
for the product of a
of a multiplicative semigroup
is
a a
for the sum of a
of an semigroup
eg
n
n
.
, ,
, ,
the capital pi production notation
a = a
the elements a , a (n )
analogous to
the capital sigma summation notation
a = a
the elements a , a (n )
additive
ii=1
n
1
2 n
ii=1
n
1
2 n
’
Â
◊ ◊ ◊
◊ ◊ ◊ Œ
+ + ◊ ◊ ◊ +
◊ ◊ ◊ Œ
2
1
2
1
P
P
thethe factorial of n
n
n factorial
n r
n n
(n )
= factorial n
=
= = n bang
= r =
= 1 2 3 = n(n
= the product of the first n positive integers
dn
rd
dfr=1
n
r=0
n
Œ
-
¥ ¥ ¥ ◊ ◊ ◊¥ - - ◊ ◊ ◊
’ ’-
P
!
( )
)( )
1
1 2 1
GG18-23
note: just as the sum of the empty set
of additive semigroup elements
is defined to be the additive identity element zero iie
ie
a = 0 iie
so analogously the product of the empty set
of multiplicative semigroup elements
is defined to be the multiplicative identity element unity iie
ie
a = 1 iie
ii
df
ii
df
Œ∆
Œ∆
Â
’
GG18-24
C the. notation n! for n factorial was introduced
in 1808 by Christian Kramp of Strasbourg, France;
in 1812 Gauss introduced capital pi
to denote continued products;
the indicial notation was added later by others;
the Greek letter pi
corresponds phonetically & in transliteration to
the Latin - English letter pee P p
which is the initial letter of productum (Latin) = product
P
Pp
GG18-25
D
the
x r
x n
the
x r
x n
.
•
( )
( )
•
( )
( )
two dual kinds of factorial powers of x
where x is an element of a commutatve unital ring
& n
rising nth factorial power of x
= x
= x rising n (factorial)
=
= x (x +1)(x + 2) which has exactly n factors
falling nth factorial power of x
= x
= x falling n (factorial)
=
= x (x 1)(x 2) which has exactly n factors
dnn
rd
dfr=0
n
dnn
rd
dfr=0
n
Œ
+
◊ ◊ ◊ + -
-
- - ◊ ◊ ◊ - +
-
-
’
’
P
1
1
1
1
GG18-26
D the three binomial formulas / theorems
for three kinds of powers
in a commutative unital ring (n )
• the binomial formula / theorem
for ordinary powers
• the binomial formula / theorem
for rising factorial powers
• the binomial formula / theorem
for falling factorial powers
Œ
+( ) = ÊËÁ
ˆ¯̃
-
+( ) = ÊËÁ
ˆ¯̃
-
+( ) = ÊËÁ
ˆ¯̃
-
=
=
=
Â
Â
Â
P
a b n n
ran r br
a b n n
ran r br
a b n n
ran r br
r
n
r
n
r
n
0
0
0 GG18-27
D three examples of infinite products
• 1 = 1
2
1 = 2
1 = 4
first
is the product of
the second product and the third product
n=2
=1
n=1
-ÊË
ˆ¯
-ÊËÁ
ˆ¯̃
-+
ÊËÁ
ˆ¯̃
•
•
•
’
’
’
1
1
2
1
2 1
2
2
2
n
n
n
note the product
n
•( )
•( )
:
p
p
GG18-28