- Use sigma notation to add up integers and their powers
Sigma Notation
To simplify writing lengthy sums, we use sigma notation (summation notation). The Greek letter [latex]Σ[/latex] represents the sum of values. For example, if we want to add all the integers from [latex]1[/latex] to [latex]20[/latex], instead of writing out
we can use sigma notation:
Typically, sigma notation is presented in the form
where [latex]a_i[/latex] are the terms being added and [latex]i[/latex] is the index.
sigma notation
Sigma notation uses the Greek letter sigma ([latex]∑[/latex]) to represent the sum of a series of terms.
[latex]\displaystyle\sum_{i=1}^{n} a_i[/latex]
Each term [latex]a_i[/latex] is evaluated for all integer values of [latex]i[/latex] from the lower limit to the upper limit, and then all these values are added together.
How to: Evaluate Sigma Notation
- Identify the Index and Limits: Locate the index variable [latex]i[/latex], the starting value (often [latex]1[/latex]), and the upper limit [latex]n[/latex].
- Determine the Term Expression: Identify the term [latex]a_i[/latex] that you will be summing.
- Evaluate Each Term: Substitute each integer value from the starting value to the upper limit into the term expression.
- Sum the Evaluated Terms: Add up all the evaluated terms to get the final sum.
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.
- Write in sigma notation and evaluate the sum of terms [latex]3^i[/latex] for [latex]i=1,2,3,4,5[/latex].
- Write the sum in sigma notation:
[latex]1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}[/latex].
The following rules summarize key properties of sigma notation.
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].
-
[latex]\underset{i=1}{\overset{n}{\Sigma}}c=nc[/latex]
-
[latex]\underset{i=1}{\overset{n}{\Sigma}}ca_i=c\underset{i=1}{\overset{n}{\Sigma}}a_i[/latex]
-
[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]
-
[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]
-
[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
Let’s prove properties 2 and 3, and leave the proof of the other properties for the examples.
Proof of Property 2:
Proof of Property 3:
[latex]_\blacksquare[/latex]
Here are a few more formulas that simplify the summation process for frequently encountered functions. These rules, which apply to sums and powers of integers, will be used in the upcoming examples.
Sums and Powers of Integers
- 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].
- 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].
- 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].
Write using sigma notation and evaluate:
- The sum of the terms [latex](i-3)^2[/latex] for [latex]i=1,2,\cdots,200[/latex].
- The sum of the terms [latex](i^3-i^2)[/latex] for [latex]i=1,2,3,4,5,6[/latex].
Find the sum of the values of [latex]4+3i[/latex] for [latex]i=1,2,\cdots,100[/latex].
Evaluate the sum indicated by the notation [latex]\displaystyle\sum_{k=1}^{20} (2k+1)[/latex].