ams/econ 11b
supplementary notes
ucsc
Summation
c
2008, Yonatan Katznelson
1.
Notation
Many problems in mathematics and its applications (e.g., statistics) involve sums
with more than two terms. Writing all of the terms in such a sum can be cumbersome,
especially if there are many terms. In some cases we can use
ellipses
, for example we
might write
1 + 2 + 3 +
· · ·
+ 100
to indicate the sum of the integers from one to one hundred. But this is a little vague,
and in many cases, it might not be clear what terms are missing.
†
The Swiss mathematician Leonard Euler (pronounced
oiler
) introduced notation
for sums, using the greek letter Σ, which is an uppercase
sigma
.
‡
Definition
:
Given the terms,
A
m
, A
m
+1
, A
m
+2
, . . . , A
n
, we denote their sum by
(1.1)
A
m
+
A
m
+1
+
A
m
+2
+
· · ·
+
A
n
=
n
X
k
=
m
A
k
.
The variable
m
is called the
lower limit of summation
,
n
is called the
upper limit
of summation
and
k
is called the
index of summation
.
In this notation it is understood that
m
and
n
are integers and
m
≤
n
. Furthermore,
the index of summation
k
increases by increments of 1, starting from
m
and ending
at
n
.
The terms
{
A
m
, . . . , A
n
}
may be a list of numbers (e.g., data from an experiment),
in which case the index
k
simply enumerates the list. On the other hand, the terms
A
k
may depend on
k
, i.e.,
A
k
may be a function of
k
— this is the case that we will
be considering here.
In all of the following examples, the terms of the sum are explicit functions of the
index of summation.
Examples.
1a.
6
X
k
=1
k
= 1 + 2 + 3 + 4 + 5 + 6.
1b.
100
X
j
=0
(2
j
) = 0 + 2 + 4 +
· · ·
+ 200.
†
Can you tell what terms are missing from the sum 1 + 5 + 11 +
· · ·
+ 109?
‡
This notation for sums is therefore also called ‘sigma notation’.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
1c.
15
X
i
=3
(
i
2
+
i

1) = 11 + 19 + 29 +
· · ·
+ 239.
Comments:
As you can see in the examples, we can use letters other than
k
to
denote the index of summation. The letters
i, j
and
k
are common choices, as are
m
and
n
(when they’re not being used to denote the limits of summation). Also, as you
can see in example
1b.
, the lower limit of summation may be zero (or negative).
Sigma notation is simply that — notation. It is not a tool for
evaluating
the sums
in question. To evaluate sums, we’ll use the basic properties of addition to develop
some simple rules and formulas. On the other hand, the Σnotation will make these
rules and formulas easier to express and understand.
2.
Basic rules.
The only operation being used in the sum
∑
n
k
=
m
A
k
is addition.
It follows that
all the basic properties of addition hold for such sums. In particular, we can rear
range the terms in a sum, we can collect terms to split a sum into smaller sums and
multiplication by a constant factor
distributes
over a sum. The following three rules
illustrate these properties.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 BINICI
 Addition, Summation, Mathematical notation

Click to edit the document details