Sumrule - ECON summation operator summary PDF

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ECON summation operator summary ...

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The Summation Operator



The Greek capital letter

(sigma) denotes the summation operator.

Let x1 , x2 , . . . , x n be a set of numbers. The sum of the numbers is written as: n

 xi i= 1 n

This gives the calculation:

 xi = x1 + x 2 + . . . + x n i =1

•

The letter i is called the index of summation. Other letters, for example j or k, may be used for the index of summation. The numbers 1 and n are the lower limit and upper limit of the summation.

•

The expression

•

n

 xi

can be stated in words as:

i= 1

“sum the numbers x i for all values of i from 1 to n”. 9

 xi = x 6 + x 7 + x 8 + x 9

Example :

i= 6

Denote b and c as constant numbers. Some useful properties of the summation operator are: n

(1)

 b = n⋅b i =1 3

Example :

 5 = ( 5 + 5 + 5) = 3 ⋅ 5 = 15 i =1

n

(2)

n

 b ⋅x i = b

x i

i =1

i =1

Results (1) and (2) can be applied to get: (3)

n

n

i=1

i =1

 ( b + c ⋅ xi ) = n ⋅ b + c  xi Summation Operator - 1

With another set of numbers y1 , y2 , . . . , yn results are: n

n

n

i =1

i =1

i =1

 (xi + yi ) =  xi +  yi

and

n

n

n

i =1

i= 1

i =1

 (b ⋅ xi + c ⋅ yi ) = b  xi + c  yi The index of summation may be the variable to be summed. 5

Example :

i = 2+ 3 + 4+ 5 i= 2

An abbreviated form of the summation notation can be used. For example, if f(x) is a function of numeric values x, the summation over all values of x can be stated as:

 f( x ) x

Example : Let f (x) = x2 . For values x = 2, 5, 7, 10 then

 f(x) = 2 ⋅ 2 + 5 ⋅ 5 + 7 ⋅ 7 + 10 ⋅ 10 = 178 x

Summation Operator - 2...