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Study Guides > College Algebra

Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.

3+7+11+15+19+..3+7+11+15+19+...
The nth n\text{th } partial sum of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation  Sn \text{ }{S}_{n}\text{ } represents the partial sum.
S1=3S2=3+7=10S3=3+7+11=21S4=3+7+11+15=36\begin{array}{l}{S}_{1}=3\\ {S}_{2}=3+7=10\\ {S}_{3}=3+7+11=21\\ {S}_{4}=3+7+11+15=36\end{array}
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma, σ\sigma, to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series. Explanation of summation notion as described in the text. If we interpret the given notation, we see that it asks us to find the sum of the terms in the series ak=2k{a}_{k}=2k for k=1k=1 through k=5k=5. We can begin by substituting the terms for kk and listing out the terms of this series.
a1=2(1)=2a2=2(2)=4a3=2(3)=6a4=2(4)=8a5=2(5)=10\begin{array}{l}\begin{array}{l}\\ {a}_{1}=2\left(1\right)=2\end{array}\hfill \\ {a}_{2}=2\left(2\right)=4\hfill \\ {a}_{3}=2\left(3\right)=6\hfill \\ {a}_{4}=2\left(4\right)=8\hfill \\ {a}_{5}=2\left(5\right)=10\hfill \end{array}
We can find the sum of the series by adding the terms:
k=152k=2+4+6+8+10=30\sum _{k=1}^{5}2k=2+4+6+8+10=30

A General Note: Summation Notation

The sum of the first nn terms of a series can be expressed in summation notation as follows:
k=1nak\sum _{k=1}^{n}{a}_{k}
This notation tells us to find the sum of ak{a}_{k} from k=1k=1 to k=nk=n. kk is called the index of summation, 1 is the lower limit of summation, and nn is the upper limit of summation.

Q & A

Does the lower limit of summation have to be 1?

No. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples with lower limits of summation other than 1.

How To: Given summation notation for a series, evaluate the value.

  1. Identify the lower limit of summation.
  2. Identify the upper limit of summation.
  3. Substitute each value of kk from the lower limit to the upper limit into the formula.
  4. Add to find the sum.

Example 1: Using Summation Notation

Evaluate k=37k2\sum _{k=3}^{7}{k}^{2}.

Solution

According to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of k2{k}^{2} from k=3k=3 to k=7k=7. We find the terms of the series by substituting k=3,4,5,6k=3\text{,}4\text{,}5\text{,}6, and 77 into the function k2{k}^{2}. We add the terms to find the sum.
k=37k2=32+42+52+62+72=9+16+25+36+49=135\begin{array}{ll}\sum _{k=3}^{7}{k}^{2}\hfill & ={3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}\hfill \\ \hfill & =9+16+25+36+49\hfill \\ \hfill & =135\hfill \end{array}

Try It 1

Evaluate k=25(3k1)\sum _{k=2}^{5}\left(3k - 1\right). Solution

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