How do you find the sum of the convergent series $12 + 4 + \dfrac{4}{3} + ...$ If the convergent series is not convergent, how do you know?

Hint: A series is defined as an expression in which infinitely many terms are added one after the other to a given starting quantity. It is represented as $\sum\limits_{n = 1}^\infty {{a_n}} $ where $\sum {} $ sign denotes the summation sign which indicates the addition of all the terms. In this question, we are given a GP and we have to find the sum of the GP. We see that the common ratio of the G.P. is $\dfrac{1}{3}$ as each term of the G.P. is $\dfrac{1}{3}$ times its previous term. We do a test called the ratio test, also known as D’Alembert’s ratio test or the Cauchy ratio to test the convergence of a series $\sum\limits_{n = 1}^\infty {{a_n}} $ .

Complete step-by-step solution:
The nth term of a G.P. is given as ${a_n} = a{r^{n - 1}}$
Now to do the ratio test, we have $L = \mathop {\lim }\limits_{x \to \infty } \left| {\dfrac{{{a_{n + 1}}}}{{{a_n}}}} \right|$
Putting the value of ${a_n}$ in the above equation, we have –
$L = \mathop {\lim }\limits_{n \to \infty } \left| {\dfrac{{a{r^n}}}{{a{r^{n - 1}}}}} \right|$
As $n \to \infty $ , the limit clearly comes to be positive, so we can remove the modulus symbol.
$
  L = \mathop {\lim }\limits_{n \to \infty } {r^{n - (n - 1)}} \\
   \Rightarrow L = \mathop {\lim }\limits_{n \to \infty } {r^{n - n + 1}} \\
   \Rightarrow L = \mathop {\lim }\limits_{n \to \infty } r \\
   \Rightarrow L = \dfrac{1}{3} \\
  as\,\dfrac{1}{3} < 1 \\
   \Rightarrow L < 1 \\
 $
Hence, the series $12 + 4 + \dfrac{4}{3} + ...$ converges by the ratio test.
We know that the sum of an infinite G.P. is –
$
  S = \dfrac{a}{{1 - r}} \\
   \Rightarrow S = \dfrac{{12}}{{1 - \dfrac{1}{3}}} = \dfrac{{12}}{{\dfrac{{3 - 1}}{3}}} = 12 \times \dfrac{3}{2} \\
   \Rightarrow S = 18 \\
 $
Hence, the sum of the convergent series $12 + 4 + \dfrac{4}{3} + ...$ is 18.

Note: When we get further and further in a sequence, the terms get closer and closer to a specific limit, this signifies the convergence of the series, and the series is said to be convergent series. A series is said to be convergent if the value of L comes out to be smaller than 1, and the series is said to be divergent if the value comes out to be greater than 1. The test is said to be inconclusive if it is equal to 1 as the limit fails to exist. Doing this test on the given series we can find out whether the series is converging or not.