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Sum Notation and Properties of Sigma

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Sigma Notation

Writing the sum of long lists of numbers that have a specific pattern is not very efficient. Summation 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?

1+4+9+16+25+\cdots+144

Watch This

http://www.youtube.com/watch?v=0L0rU17hHuM&feature=youtu.be James Sousa: Find a Sum Written in Summation/Sigma Notation

Guidance

A series is a sum of a sequence. The Greek capital letter sigma is used for summation notation because it stands for the letter  S as in sum.

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

a_1, a_2, a_3, a_4, a_5

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.

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

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, a_i , tells you what terms are added together. The lower index, i=1 , 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.

Example A

Write out all the terms of the series.

\sum\limits_{k=4}^8 2k

Solution:

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

Example B

Write the sum in sigma notation: 2+3+4+5+6+7+8+9+10

Solution:

2+3+4+5+6+7+8+9+10=\sum\limits_{i=2}^{10} i

Example C

Write the sum in sigma notation.

\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}

Solution:

\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}

Concept Problem Revisited

The hardest part when first using sigma representation is determining how each pattern generalizes to the  k^{th} term. Once you know the k^{th} term, you know the argument of the sigma. For the sequence creating the series below, a_k=k^2 . Therefore, the argument of the sigma is i^2 .

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

Vocabulary

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.

Guided Practice

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

\sum\limits_{k=0}^4 3k-1

2. Represent the following infinite series in summation notation.

\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots

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

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

Answers:

1. \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

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

\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}

3. Just like summation uses a capital Greek letter for S , product uses a capital Greek letter for  P which is the capital form of \pi .

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)

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.

Practice

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

1. \sum\limits_{k=1}^5 2k-3

2.  \sum\limits_{k=0}^8 2^k

3. \sum\limits_{i=1}^4 2 \cdot 3^i

4. \sum\limits_{i=1}^{10} 4i-1

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

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

6. 1+4+7+10+13+16+19+22

7.  3+5+7+9+11

8. 8+7+6+5+4+3+2+1

9. 5+6+7+8

10. 3+6+12+24+48+\cdots

11. 10+5+\frac{5}{2}+\frac{5}{4}

12. 4-8+16-32+64\cdots

13. 2+4+6+8+\cdots

14. \frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\cdots

15. \frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\frac{2}{81}+\cdots

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