Question

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.

Answer 1

A geometric series is the sum of the terms of a geometric sequence of  the form $a, ar, ar^{2}, ar^{3},...$, 
where r is the common ratio, and a≠0 is the first term.

That is, the series is  $a+ar+ar^{2}+ar^{3},...$

In sigma notation, the series is written as:

∞
∑  $a r^{k}$ 
k=0
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The geometric series of the form
∞
∑  $a r^{k}$, converges to $\frac{a}{1-r}$ 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:
$\frac{a}{1-r}=\frac{940}{1-1/5}=\frac{940}{4/5}=752$


Answer: 

∞          
∑  $940 (1/5)^{k}$=752   ( the upper limit of the population is 752)
k=0