Sigma Notation and Series
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Consider the sequence 10.20, 11.40, 12.10, 13.40 where each term represents the amount of money you earned as interest on your savings account for each of four years.

The sum of the terms, 10.2 + 11.4 + 12.1 + 13.4, represents the total interest you earned in the four year period.  Such a sequence summation is called a series and is designated by Sn where n represents the number of terms of the sequence being added.

Sn is often called an nth partial sum, since it can represent
the sum of a certain "part" of a sequence.

A series can be represented in a compact form,
called summation notation, or sigma notation.
The Greek capital letter sigma, ,
 is used to indicate a sum.

"The summation from 1 to 4 of 3n":

 

 

Examples:

Problem: Answer:
1.  Evaluate:  

 

2.  Evaluate: 

Notice how only the variable i was replaced with the values from 2 to 4.

 

3.  Evaluate:
    

Notice how raising (-1) to a power affected the signs of the terms.  This is an important pattern strategy to remember.

 

4.  Evaluate:   While the starting value is usually 1, it can actually be any integer value.

 

5.  Use sigma notation to
             represent
        2 + 4 + 6 + 8 + ...
           for 45 terms.

 

Look for a pattern based upon the position of each term.  In this problem, each term is its position location times 2, for a sequence formula of
           
term position term
1 2
2 4
3 6
n 2n

One possible answer:

6.  Use sigma notation to
            represent
     -3 + 6 - 9 + 12 - 15 + ...
           for 50 terms.
Again, look for a pattern.  Each term is its position location times 3, but with signs alternating.  Example #3 showed how to create alternating signs using powers of -1.
term position term
1 -3
2 6
3 -9
4 12

One possible answer:


Strategies to remember when trying to find an expression for a sequence (series):

Example Possible notation
(partial sum)
Strategy
Look to see if a value is being consistently added (or subtracted)
OR  Be aware that there is more than one answer.
Patterns can increase or decrease.
Look to see if a value is being consistently multiplied (or divided)
Look to see if the values are "famous" numbers such as perfect squares.
Look to see if the signs alternate.  Alternating signs can be handled using powers of -1.

 

Check out how to use your TI-83+/84+ graphing calculator with summation notation. Click here.