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supplementary notes

ams/econ 11b

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 toindicate 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 upper-case sigma.‡ Definition: Given the terms, Am , Am+1 , Am+2 , . . . , An , we denote their sum by
n

(1.1)

Am +Am+1 + Am+2 + · · · + An =
k=m

Ak .

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 {Am , . . . , An } may be a list ofnumbers (e.g., data from an experiment), in which case the index k simply enumerates the list. On the other hand, the terms Ak may depend on k, i.e., Ak 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.
6

1a.
k=1 100

k = 1 + 2 + 3 + 4 + 5 + 6. (2j) = 0 + 2+ 4 + · · · + 200.
j=0

1b.
† ‡

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

2
15

1c.
i=3

(i2 + 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 commonchoices, 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, theΣ-notation will make these rules and formulas easier to express and understand.

2.

Basic rules.

The only operation being used in the sum n Ak is addition. It follows that k=m all the basic properties of addition hold for such sums. In particular, we can rearrange the terms in a sum, we can collect terms to split a sum into smaller sums and multiplication by a constant factor distributesover a sum. The following three rules illustrate these properties.
n n n

R1. Rearranging terms: R2. Collecting terms:
k=m

(Ak + Bk ) =
k=m n l k=m n

Ak Ak .
k=l+1 n

+
k=m

Bk

.

Ak =
k=m n

Ak +

R3. Distributive property:
k=m

cAk = c
k=m

Ak

.

Examples.
6 3 6

2a.
k=1 100

k=
k=1

k+
k=4 100

k. j .

2b.
j=0 15

(2j) = 2
j=0

15

1515

2c.
i=3

(i + i − 1) =
i=3

2

i +
i=3

2

i+
i=3

(−1).

Comments: Rules 1. and 2. rely on the associative and commutative properties of addition. Also, as you can see in the last sum on the right of Example 2c., the terms in a sum may also be constant.

3

3.

Simple formulas.

One way of evaluating a sum is to simply add all of the terms together, one afteranother, until we’re done. This is easy to do when the number of terms is small, e.g.,
6

k = 1+2+3+4+5+6 = 3+3+4+5+6 = 6+4+5+6 = 10+5+6 = 15+6 = 21,
k=1

and in many cases this may be the only option (for example in statistical applications). On the other hand, there are many sums that can be evaluated using formulas. In this section, I’ll introduce two simple formulas, and in the next two...