Virtual University Course MTH101 - Calculus and Analytical Geometry Lecture No 27 Instructor's Name: Dr. Faisal Shah Khan Course Email: mth101@vu.edu.pk

1. 27-Error: Reference source not found
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Lecture No. 27 Sigma Notation
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Similarly if f(k) is a function of k, and a and b
are integers such that a b≤ , then
( )
b
k a
f k
=
∑
denote the sum of the terms that result when we
substitute successive Integers for k, stating
with k=a and ending with k=b i.e.
Here are some of its examples.
4 4 4
2 2 2
1 1 1
2 2 2 2
,
1 2 3 4
i m p
i m and p denote the same
sum
= = =
= + + +
∑ ∑ ∑
Thus

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2 1
3 3
2 1
1 1 1
2
2 1 2 3i i
k and
i= =
= = =
+ +
∑ ∑
If in the expression to right of does not
involve the index of summation, then we take
all the terms in the same to be the same, with
each allowable value of the summation index.
Thus,
Changing the Index of Summation
On occasion we shall want to change
the sigma notion for a given sum to a sigma not-
-ation with different limits of summation. The
following example illustrate a method for doing
this.
If we want to represent the sum as general
sum, then general sum for three and n terms
are;

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Example
( ) ( )
( )
( ) ( )( ) ( )
( ) ( )
( ) ( ) ( )
30 30
2
1 1
30 30
2
1 1
1
{ 5.4.1
30 30 1 2 30 1 30 30 1
6 2
{ 5.4.2
30 31 61 30 31
9920
6 2
k k
k k
k k k k
k k
by a
by a and b
= =
= =
+ = +
= +
+ + +
= +
= + =
∑ ∑
∑ ∑