Explain how to tell if a geometric series is convergent or divergent. include an example of a convergent series and an example of a divergent series in your explanation.

The geometric series is where the ratio of consecutive terms  is same.

We know that general representation of geometric series is:

So, we say a given geometric series is :

Convergent if |r|<1  and converges at  a/1-r

Diverges if |r|>=1

Example:

We have a series:  2,4,8,16

Here the common ratio of the above series is :

Formula for common ratio:

Here,

So, the common ratio is 2> 1 so, series diverges.

Now, we have a series

16,8,4,2...

So, here r=1/2<1

Converging point is: a/1-r

Here, a=16, r=1/2 on substituting the values we get:

.