Question

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.

Answer 1

Answer:

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

We know that general representation of geometric series is:

$a,ar^2,ar^3....$

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: $\frac{a_2}{a_1},\frac{a_3}{a_2}..$

Here, $r=\frac{4}{2}=2$

$r=\frac{8}{4}=2$

$r=\frac{16}{8}=2$

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

Now, we have a series

16,8,4,2...

$r=\frac{8}{16}=\frac{1}{2}$

$r=\frac{4}{8}=\frac{1}{2}$

$r=\frac{2}{4}=\frac{1}{2}$

So, here r=1/2<1

Converging point is: a/1-r

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

$\frac{16}{1-\frac{1}{2}}$

$\frac{16}{\frac{1}{2}}=32$.