Sigma Notation

Learning Outcomes

  • Use sigma (summation) notation to calculate sums and powers of integers

As mentioned, we will use shapes of known area to approximate the area of an irregular region bounded by curves. This process often requires adding up long strings of numbers. To make it easier to write down these lengthy sums, we look at some new notation here, called sigma notation (also known as summation notation). The Greek capital letter [latex]\Sigma[/latex], sigma, is used to express long sums of values in a compact form. For example, if we want to add all the integers from 1 to 20 without sigma notation, we have to write

[latex]1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20[/latex].

 

We could probably skip writing a couple of terms and write

[latex]1+2+3+4+\cdots+19+20[/latex],

which is better, but still cumbersome. With sigma notation, we write this sum as

[latex]\displaystyle\sum_{i=1}^{20} i[/latex],

 

which is much more compact.

Typically, sigma notation is presented in the form

[latex]\displaystyle\sum_{i=1}^{n} a_i[/latex]

 

where [latex]a_i[/latex] describes the terms to be added, and the [latex]i[/latex] is called the index. Each term is evaluated, then we sum all the values, beginning with the value when [latex]i=1[/latex] and ending with the value when [latex]i=n[/latex]. For example, an expression like [latex]\displaystyle\sum_{i=2}^{7} s_i[/latex] is interpreted as [latex]s_2+s_3+s_4+s_5+s_6+s_7[/latex]. Note that the index is used only to keep track of the terms to be added; it does not factor into the calculation of the sum itself. The index is therefore called a dummy variable. We can use any letter we like for the index. Typically, mathematicians use [latex]i[/latex], [latex]j[/latex], [latex]k[/latex], [latex]m[/latex], and [latex]n[/latex] for indices.

Let’s try a couple of examples of using sigma notation.

Example: Using Sigma Notation

  1. Write in sigma notation and evaluate the sum of terms [latex]3^i[/latex] for [latex]i=1,2,3,4,5[/latex].
  2. Write the sum in sigma notation:
    [latex]1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}[/latex].

Watch the following video to see the worked solution to Example: Using Sigma Notation.

Try It

Write in sigma notation and evaluate the sum of terms [latex]2^i[/latex] for [latex]i=3,4,5,6[/latex].

The properties associated with the summation process are given in the following rule.

Properties of Sigma Notation

Let [latex]a_1,a_2, \cdots,a_n[/latex] and [latex]b_1,b_2,\cdots,b_n[/latex] represent two sequences of terms and let [latex]c[/latex] be a constant. The following properties hold for all positive integers [latex]n[/latex] and for integers [latex]m[/latex], with [latex]1\le m\le n[/latex].

  1. [latex]\underset{i=1}{\overset{n}{\Sigma}}c=nc[/latex]
  2. [latex]\underset{i=1}{\overset{n}{\Sigma}}ca_i=c\underset{i=1}{\overset{n}{\Sigma}}a_i[/latex]
  3. [latex]\underset{i=1}{\overset{n}{\Sigma}}(a_i+b_i)=\underset{i=1}{\overset{n}{\Sigma}}a_i+\underset{i=1}{\overset{n}{\Sigma}}b_i[/latex]
  4. [latex]\underset{i=1}{\overset{n}{\Sigma}}(a_i-b_i)=\underset{i=1}{\overset{n}{\Sigma}}a_i-\underset{i=1}{\overset{n}{\Sigma}}b_i[/latex]
  5. [latex]\underset{i=1}{\overset{n}{\Sigma}}a_i=\underset{i=1}{\overset{m}{\Sigma}}a_i+\underset{i=m+1}{\overset{n}{\Sigma}}a_i[/latex]

Proof

We prove properties 2 and 3 here, and leave proof of the other properties to the Exercises.

Property 2: We have

[latex]\begin{array}{ll}\displaystyle\sum_{i=1}^{n} ca_i & =ca_1+ca_2+ca_3+\cdots+ca_n \\ & =c(a_1+a_2+a_3+\cdots+a_n) \\ & =c\displaystyle\sum_{i=1}^{n} a_i \end{array}[/latex]

 

Property 3: We have

[latex]\begin{array}{ll}\displaystyle\sum_{i=1}^{n} (a_i+b_i) & =(a_1+b_1)+(a_2+b_2)+(a_3+b_3)+\cdots+(a_n+b_n) \\ & =(a_1+a_2+a_3+\cdots+a_n)+(b_1+b_2+b_3+\cdots+b_n) \\ & =\displaystyle\sum_{i=1}^{n} a_i+ \displaystyle\sum_{i=1}^{n} b_i \end{array}[/latex]

[latex]_\blacksquare[/latex]

A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers, and we use them in the next set of examples.

Sums and Powers of Integers

  1. The sum of [latex]n[/latex] integers is given by
    [latex]\underset{i=1}{\overset{n}{\Sigma}}i=1+2+\cdots+n=\frac{n(n+1)}{2}[/latex].
  2. The sum of consecutive integers squared is given by
    [latex]\underset{i=1}{\overset{n}{\Sigma}}i^2=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}[/latex].
  3. The sum of consecutive integers cubed is given by
    [latex]\underset{i=1}{\overset{n}{\Sigma}}i^3=1^3+2^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}[/latex].

Example: Evaluation Using Sigma Notation

Write using sigma notation and evaluate:

  1. The sum of the terms [latex](i-3)^2[/latex] for [latex]i=1,2,\cdots,200[/latex].
  2. The sum of the terms [latex](i^3-i^2)[/latex] for [latex]i=1,2,3,4,5,6[/latex].

Watch the following video to see the worked solution to Example: Evaluation Using Sigma Notation.

Try It

Find the sum of the values of [latex]4+3i[/latex] for [latex]i=1,2,\cdots,100[/latex].

Example: Finding the Sum of the Function Values

Find the sum of the values of [latex]f(x)=x^3[/latex] over the integers [latex]1,2,3,\cdots,10[/latex].

Try It

Evaluate the sum indicated by the notation [latex]\displaystyle\sum_{k=1}^{20} (2k+1)[/latex].

Try It