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