9.1 Sequences and Series (continued)

Summation Notation

Summation notation, also known as sigma notation is a convenient notation for the sum of the terms of a finite sequence.

Definition: The sum of the first n terms of a sequence is represented by:

where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Example 1

               = 3 + 6 + 9 + 12
               = 30

Example 2

               = 6 + 11 + 18 + 27
               = 62

Example 3 (with factorials)

               = 2 + 1 + 1/3 + 1/12
               = 41/12


Properties of Sums
-c is any constant

1. 
2. 
3. 

Series

The sum of the terms of an infinite sequence is called an infinite series or just a series.

Definition:
Considering the infinite sequence .

1. The sum of all terms of the infinite sequence is called an infinite series and is denoted by:
2. The sum of the first n terms of the sequence is called a finite sequence or the nth partial sum.

Partial Sum Examples

1. For the series  find (a) the 3rd partial sum and (b) the sum
   a) 3rd partial sum
       
      = .3 + .03 + .003 = .333
   b) the sum of the series
       
      = .3 + .03 + .003 + .0003 + .00003 + ...
      = .33333... = 1/3