The population of a local species of flies can be found using an infinite geometric series where a1 = 940 and the common ratio is one fifth. write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

A geometric series is the sum of the terms of a geometric sequence of  the form , 
where r is the common ratio, and a≠0 is the first term.

That is, the series is  

In sigma notation, the series is written as:

∞
∑   
k=0
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The geometric series of the form
∞
∑  , converges to  if |r|<1
k=0

and diverges otherwise.
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in our problem, a , the first term is equal to 940, and the common ratio is |1/5|<1, 

thus the series converges to:


 

∞          
∑  =752   ( the upper limit of the population is 752)
k=0