Answer
series converges
and
$S_{\infty}= \dfrac{4}{7}$
Work Step by Step
The common ratio of a geometric sequence is equal to the quotient of any term and the term before it:
$ \ r = \dfrac{a_n}{a_{n-1}}$ or, $r=\dfrac{a_2}{a_1}$
The sum of a convergent infinite geometric series is given by the formula:
$S_{\infty}=\dfrac{a_1}{1-r}$ and a geometric series converges if $|r| \lt 1$.
where $r$=common ratio and $a_1$= the first term
Now, $r=\dfrac{a_2}{a_1} = \dfrac{-3/4}{2}=\dfrac{-3}{4}$
Since $|\dfrac{-3}{4}|=\dfrac{3}{4}\lt 1$, so the infinite geometric series converges.
Next, we will find the sum of the infinite geometric series when $a_1 = 1$ and $r=\dfrac{-3}{4}$
$S_{\infty} = \dfrac{a_1}{1-r} = \dfrac{1}{1-\dfrac{-3}{4}}=\dfrac{1}{7/4}$
Therefore, the sum of the convergent infinite geometric series is: $S_{\infty}= \dfrac{4}{7}$
