Series

Definition of a Series

Example

Consider the sequence

        an  =  n2  

        {1, 4, 9, 16, 25, ...} 

We make the following definition.  

        S1  = a1  =  1

        S2  =  a1 + a2  =  1 + 5  =  5

        S3  =  a1 + a2 + a3  =  1 + 4 + 9  =  13

 


Exercise 

Find S4  and S5 

S4 = 29    and    S5 = 54

 


In general., for a sequence {an}  we define a new sequence called the sequence of partial sums by

        Sn =  a1 + a2 + a3 + ... + an  

 

Exercise

Find S5 for

  1. an  =  2n + 1 
    35

  2. an  =  (-1)n  
    -1


Sigma Notation

Instead of using the " ... " notation we use the following notation:

Example

=  1/3 + 1/4 + 1/5 + 1/6

We read this as

 "The sum from i equals 3 to 6 of 1 over i." 

 i is called the index of summation.  Think of sigma as a big plus sign.  The bottom number tells you where to start and the top number tells you where to end.

 


Example

= (1 + 4) + (1 + 9) + (1 + 16) + ( 1 + 25) + (1 + 36)

 


Example

Write 

        3 + 5 + 7 + 9 + ... + 23

in sigma notation

 

Solution

  • Step 1  Identify an  (this goes to the right of the sigma sign)

    We see that 

            an  =  2n + 1

  • Step 2  Solve for n to find out what n the first term uses.  (this goes on the bottom of the sigma sign)

    Since 

            2n+ 1 = 3  

    has solution 

            n = 1

    the first term uses n = 1


  • Step 3  Find out what n the last term uses (this goes on the top of the sigma sign)

    We solve: 

            2n + 1 = 23 

    has solution 

            n = 12

  • Step 4  Write

     

           

     

    for the example,

           


Exercises

Write the following in sigma notation

 

  1. {3, 6, 9, 12, ... 120}

  2. {-1, 2, -4, ..., 128}

 


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