Infinite Geometric Series

1. Definition of an Infinite Geometric Series

   We learned that a geometric series has the form

   

   
   

   Definition  The series            is called the infinite geometric series.

   
   
   

2. Calculating the Infinite Geometric Series

    

   Example
   
   Suppose that a runner begins on a one mile track.  In the first part of the race the runner runs 1/2 of the track.  In the second part of the race the runner runs half the remaining distance (1/4 miles).  In the third part of the race the runner runs half the remaining distance (1/8 miles).  The runner continues the process indefinitely.  The runner will never complete the entire track, but after each part the runner will get closer and closer to finishing.  We say that the runner will run the entire track "in the limit".  The equation for this is
   
            
   
   We say that this geometric series converges to 1.  

   
   
   Example
   
   Suppose that there is a geometric sequence with
   
           r = 2 and a₁ = 1
   
   then the infinite geometric series is
   
           2 + 4 + 8 + 16 + ...
   
   We see that these numbers just keep increasing to infinity.  In general, if |r| > 1 then the geometric series is never defined.  We say that the series diverges.  If |r| < 1 then recall that the finite geometric series has the formula
   
           S _{i = 1}ⁿ  a₁r^i-1 = a₁[(1 - rⁿ)/(1 - r)]
   
   If n is large then rⁿ will converge asymptotically to 0 and hence we have the formula 

   
             

    

   Example:  
   
   Find 
   
           S _{i = 1} ⁿ [2(1/3)^i-1]
   

   Solution:

   We have 
   
           2/(1 - 1/3) = 2/(2/3) = 3

    

    

3. Repeating Decimals
   
   Recall that a rational number in decimal form is defined as a number such that the digits repeat.  We can use a geometric series to find the fraction that corresponds to a repeating decimal.

    

   Example:
   
           .737373737... = .73 + .0073 + .000073 + ...
   
   we have 
   
           a₁ = .73     and     r = .01 
   
   Hence
   
           .73737373.... = .73/(1 - .01) = (73/100)/(99/100) = 73/99

    

   Example:
   
           4.16826826826826... =  4.1 + .1[.682 + .000682 + .000000682 + ...]
   
   we have 
   
           a₁ = .682 and r = .001 
   
   Hence
   
           4.16826826826826... =  4.1 + .1[.682/(1 - .001) 
   
           = 41/10 + (1/10)(682/1000)/(999/1000)
   
           = 41/10 + 682/9990 = 41,641/9990

   
   

4. Interval Of Convergence 
   
   If an infinite series involves a variable x, then we call the interval of convergence the set of all x such that the interval converges for that x.  For example
   
   
            
   
   has interval of convergence
   
           -1 < x < 1.

    

   Example:  
   
   Find the interval of convergence of

   

    

   Solution
   
   We write 
   
           |3x - 2| < 1 
   
   we solve
   
           3x - 2 =1 or 3x - 2 = -1
   
   Adding 2 and dividing by three gives
   
           x = 1     or     x = 1/3
   
   Hence     
   
           1/3 < x < 1 
   
   is the interval of convergence.

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