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:

.

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752