Answer:
The geometric series is where the ratio of consecutive terms is same.
We know that general representation of geometric series is:
[tex]a,ar^2,ar^3....[/tex]
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: [tex]\frac{a_2}{a_1},\frac{a_3}{a_2}..[/tex]
Here, [tex]r=\frac{4}{2}=2[/tex]
[tex]r=\frac{8}{4}=2[/tex]
[tex]r=\frac{16}{8}=2[/tex]
So, the common ratio is 2> 1 so, series diverges.
Now, we have a series
16,8,4,2...
[tex]r=\frac{8}{16}=\frac{1}{2}[/tex]
[tex]r=\frac{4}{8}=\frac{1}{2}[/tex]
[tex]r=\frac{2}{4}=\frac{1}{2}[/tex]
So, here r=1/2<1
Converging point is: a/1-r
Here, a=16, r=1/2 on substituting the values we get:
[tex]\frac{16}{1-\frac{1}{2}}[/tex]
[tex]\frac{16}{\frac{1}{2}}=32[/tex].
