Series

Definition of a Series

Example

Consider the sequence

        a_n  =  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 

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

        S_n =  a₁ + a₂ + a₃ + ... + a_n  

Exercise

Find S₅ for

1. a_n  =  2n + 1 
   
   
   

2. a_n  =  (-1)ⁿ  
   

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 a_n  (this goes to the right of the sigma sign)
  

  We see that 

          a_n  =  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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