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?

Feb 15, 2016

See explanation.

Explanation:

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

$q = \frac{4}{12} = \frac{1}{3}$

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

$S = {a}_{1} / \left(1 - q\right)$

$S = \frac{12}{1 - \frac{1}{3}} = \frac{12}{\frac{2}{3}} = 12 \cdot \frac{3}{2} = 18$