Chapter 9: Sequences, Sums & Products

This chapter introduces notation for working with ordered lists of numbers and with repeated addition or multiplication. These ideas appear throughout mathematics whenever we want to describe a pattern compactly instead of writing out every term.

Sequences

A sequence is an ordered list of numbers. For example, the numbers

$$0,1,2,3,4,5,\dots$$

form a sequence. This is different from a set because, in a sequence, the order of the terms matters. For this reason, when we use variables to represent terms in a sequence, we attach an index to each term:

$$a_0,a_1,a_2,a_3,\dots$$

The numbers in the subscripts are called indices (the plural of index).

Summation Notation

Given a sequence $\{a_n{\}}_{n=k}^{\infty }$ and numbers $m$ and $p$ satisfying $k\le m\le p$, the summation from $m$ to $p$ of the sequence $\{a_n\}$ is written

$$\sum _{n=m}^p{a}_n=a_m+a_{m+1}+\dots +a_p.$$

The variable $n$ is called the index of summation. The number $m$ is called the lower limit of summation, while the number $p$ is called the upper limit of summation.

The index variable is considered a "dummy variable" in the sense that it may be changed to any letter without affecting the value of the summation. For instance,

$$\sum _{n=3}^6(2n-1)=\sum _{k=3}^6(2k-1)=\sum _{j=3}^6(2j-1).$$

One place you may encounter summation notation is in mathematical definitions. For example, summation notation allows us to define polynomials as functions of the form:

$$\begin{array}{rl}f(x)& =a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0\\ & =\sum _{k=0}^n{a}_k{x}^k.\end{array}$$

Here:

• $n$ is a non-negative integer (the degree of the polynomial).
• $a_n,a_{n-1},\dots ,a_0$ are real constants.
• $a_n\ne 0$ if $n>0$.

Product Notation

If we want to multiply elements of a sequence instead of adding them, we use product notation.

Given a sequence $\{a_n{\}}_{n=k}^{\infty }$ and numbers $m$ and $p$ satisfying $k\le m\le p$, the product from $m$ to $p$ of the sequence $\{a_n\}$ is written

$$\prod _{n=m}^p{a}_n=a_m\cdot a_{m+1}\cdot \dots \cdot a_p.$$

Again, the variable $n$ is the index. The number $m$ is the lower limit while the number $p$ is the upper limit.

Chapter Exercises