Question

How do you find the sum of the convergent series 12+4+4/3+.... If the convergent series is not convergent, how do you know?

Answer 1

See explanation.

Explanation:

To find if a geometric series is convergent you have to calculate the quotient of 2 consecutive terms of the sequence `(q=a_{n+1}/a_n)`. In this case the quotient is:

`q=4/12=1/3`

If the quotient is between `-1` and `1` then the sequence in convergent and you can calculate the sum of all terms as:

`S=a_1/(1-q)`

`S=12/(1-1/3)=12/(2/3)=12*3/2=18`