Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$4 + 3 + \frac{9}{4} + \frac{27}{16} + \dots \dots \dots \dots \dots$

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Solution

Finding the sum of the given geometric series :
Given, $4 + 3 + \frac{9}{4} + \frac{27}{16} + \dots \dots \dots \dots \dots$
First term, $a = 4$
Common ratio $(r) = \frac{3}{4} (r = \frac{a_{2}}{a_{1}})$
Since, $r < 1$ . Hence the series is convergent.
As we know, the sum of a convergent series $r < 1,$ is
$S_{\infty} = \frac{a}{1 - r}$
Put the values in the above equation,
Sum of all terms $= \frac{4}{(1 - \frac{3}{4})}$
$= \frac{4}{(\frac{1}{4})} = 4 \times 4 = 16$
Hence, the sum of the geometric series is $16$ .

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