Question

For a Geometric Series explain how you can tell if the series converges or diverges as you keep adding an infinite number of terms to the series. Be sure that your explanation is clear on what it means to converge or diverge. Show using an example(s).

Answer 1

It will converge if the common factor  r  is between 0 and 1.
For example:-
 A GS with  first term a1 = 2  and d = 1/2,   the  terms are:-

2 , 1 , 1/2, 1/4, 1/8,1/16 and so on

Each term gets smaller and smaller and they will become very small - close to 0.   The sum will converge to a certain value also.
In this case it will converge to  2 /( 1 - 1/2) = 4

If d > 1  the terms will increase without bounds and the sum gets bigger and bigger . The series  diverges.

for example  2, 8, 32, 128, 512 ....  where a1 = 2 and d = 4.