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Sum Notation and Properties of Sigma

Identify and state the sum of terms in finite series

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Sigma Notation

Writing the sum of long lists of numbers that have a specific pattern is not very efficient. Summation or sigma notation allows you to use the pattern and the number of terms to represent the same sum in a much more concise way. How can you use sigma notation to represent the following sum?

\begin{align*}1+4+9+16+25+\cdots+144\end{align*}

Series in Sigma Notation

A series is a sum of a sequence. Summation notation is also known as sigma notation and is a way to represent a series. It is especially useful when the series would take too long to write out without abbreviation.The Greek capital letter sigma, \begin{align*}\sum_{}^{}{}\end{align*} is used for summation notation because it stands for the letter \begin{align*}S\end{align*} as in sum. 

Consider the following general sequence and note that the subscript for each term is an index telling you the term number.

\begin{align*}a_1, a_2, a_3, a_4, a_5\end{align*}

When you write the sum of this sequence in a series, it can be represented as a sum of each individual term or abbreviated using a capital sigma.

\begin{align*}a_1+a_2+a_3+a_4+a_5=\sum\limits_{i=1}^5 a_i\end{align*}

The three parts of sigma notation that you need to be able to read are the argument, the lower index and the upper index. The argument, \begin{align*}a_i\end{align*}, tells you what terms are added together. The lower index, \begin{align*}i=1\end{align*}, tells you where to start and the upper index, 5, tells you where to end. You should practice reading and understanding sigma notation because it is used heavily in Calculus.

Take the series:

\begin{align*}\sum\limits_{k=4}^8 2k\end{align*}

Written out, this would be: 

\begin{align*}\sum\limits_{k=4}^8 2k=2\cdot4+2\cdot5+2\cdot6+2\cdot7+2\cdot8\end{align*}

    

Examples

Example 1

Earlier, you were asked how to write the sum \begin{align*}1+4+9+16+25+\cdots+144\end{align*} in sigma notation. The hardest part when first using sigma representation is determining how each pattern generalizes to the \begin{align*}k^{th}\end{align*} term. Once you know the \begin{align*}k^{th}\end{align*} term, you know the argument of the sigma. For the sequence creating the series below, \begin{align*}a_k=k^2\end{align*}. Therefore, the argument of the sigma is \begin{align*}i^2\end{align*}.

\begin{align*}1+4+9+16+25+\cdots+144=1^2+2^2+3^2+4^2+\cdots 12^2=\sum\limits_{i=1}^{12} i^2\end{align*}

Example 2

Write the sum in sigma notation.

\begin{align*}\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}\end{align*}

\begin{align*}\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}=\sum\limits_{i=1}^7 \frac{1}{i^2}\end{align*}

Example 3

Write out all the terms of the sigma notation and then calculate the sum.

\begin{align*}\sum\limits_{k=0}^4 3k-1\end{align*}

\begin{align*}\sum\limits_{k=0}^4 3k-1&=(3\cdot0-1)+(3\cdot1-1)+(3\cdot2-1)+(3\cdot3-1)+(3\cdot4-1)\\ &=-1+2+5+8+11\end{align*}

Example 4

Represent the following infinite series in summation notation.

\begin{align*}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots\end{align*}

There are an infinite number of terms in the series so using an infinity symbol in the upper limit of the sigma is appropriate.

\begin{align*}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots=\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots=\sum\limits_{i=1}^\infty \frac{1}{2^i}\end{align*}

Example 5

Is there a way to represent an infinite product? How would you represent the following product?

\begin{align*}1\cdot\sin\left(\frac{360}{3}\right)\cdot\sin\left(\frac{360}{4}\right)\cdot\sin\left(\frac{360}{5}\right)\cdot\sin\left(\frac{360}{6}\right)\cdot\sin\left(\frac{360}{7}\right)\cdot\ldots\end{align*}

Just like summation uses a capital Greek letter for \begin{align*}S\end{align*}, product uses a capital Greek letter for \begin{align*}P\end{align*} which is the capital form of \begin{align*}\pi\end{align*}.

\begin{align*}1\cdot\sin\left(\frac{360}{2\cdot3}\right)\cdot\sin\left(\frac{360}{2\cdot4}\right)\cdot\sin\left(\frac{360}{2\cdot5}\right)\cdot\sin\left(\frac{360}{2\cdot6}\right)\cdot\sin\left(\frac{360}{2\cdot7}\right)\cdot\ldots=\prod \limits_{i=3}^{\infty} \sin \left(\frac{360}{2 \cdot i} \right)\end{align*}

This infinite product is the result of starting with a circle of radius 1 and inscribing a regular triangle inside the circle. Then you inscribe a circle inside the triangle and a square inside the new circle. The shapes alternate being inscribed within each other as they are nested inwards: circle, triangle, circle, square, circle, pentagon, ... The question that this calculation starts to answer is whether this process reduces to a number or to zero.

Review

For 1-5, write out all the terms of the sigma notation and then calculate the sum.

1. \begin{align*}\sum\limits_{k=1}^5 2k-3\end{align*}

2. \begin{align*} \sum\limits_{k=0}^8 2^k\end{align*}

3. \begin{align*}\sum\limits_{i=1}^4 2 \cdot 3^i\end{align*}

4. \begin{align*}\sum\limits_{i=1}^{10} 4i-1\end{align*}

5. \begin{align*}\sum\limits_{i=0}^4 2\cdot\left(\frac{1}{3}\right)^i\end{align*}

Represent the following series in summation notation with a lower index of 0.

6. \begin{align*}1+4+7+10+13+16+19+22\end{align*}

7. \begin{align*} 3+5+7+9+11\end{align*}

8. \begin{align*}8+7+6+5+4+3+2+1\end{align*}

9. \begin{align*}5+6+7+8\end{align*}

10. \begin{align*}3+6+12+24+48+\cdots\end{align*}

11. \begin{align*}10+5+\frac{5}{2}+\frac{5}{4}\end{align*}

12. \begin{align*}4-8+16-32+64\cdots\end{align*}

13. \begin{align*}2+4+6+8+\cdots\end{align*}

14. \begin{align*}\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots\end{align*}

15. \begin{align*}\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\cdots\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.3. 

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Vocabulary

arithmetic series

An arithmetic series is the sum of an arithmetic sequence, a sequence with a common difference between each two consecutive terms.

geometric series

A geometric series is a geometric sequence written as an uncalculated sum of terms.

sequence

A sequence is an ordered list of numbers or objects.

series

A series is the sum of the terms of a sequence.

sigma notation

Sigma notation is also known as summation notation and is a way to represent a sum of numbers. It is especially useful when the numbers have a specific pattern or would take too long to write out without abbreviation.

summation

Sigma notation is also known as summation notation and is a way to represent a sum of numbers. It is especially useful when the numbers have a specific pattern or would take too long to write out without abbreviation.

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