How do you find the sum of infinite geometric series \[5 + \dfrac{5}{3} + \dfrac{5}{9} + \dfrac{5}{{27}} + ......\] ?

Hint: From the given geometric series take the common terms out and identify the first term of the series ${a_1}$ and common ratio $r$. Then confirm that $r$ is $ - 1 < r < 1$. Substitute values for ${a_1}$ and $r$ into the formula, $S = \dfrac{{{a_1}}}{{1 - r}}$ and substitute the values to find S.

Complete step-by-step solution:
The given geometric series is
\[ \Rightarrow 5 + \dfrac{5}{3} + \dfrac{5}{9} + \dfrac{5}{{27}} + ......\]
Now let us take the same term as common
We get,
$ \Rightarrow 5\left( {1 + \dfrac{1}{3} + \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} + .....} \right)$
Now the sum $1 + \dfrac{1}{3} + \dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} + .....$ is given by the formula
$S = \dfrac{{{a_1}}}{{1 - r}}$
Where ${a_1}$ is the first term of the series and $r$ is the ratio of successive terms .
In our case $r = \dfrac{1}{3}$
Hence substitute the values in the formula We get,
$ \Rightarrow S = \dfrac{1}{{1 - \dfrac{1}{3}}}$
On resolving the fraction, we get
$ \Rightarrow S = \dfrac{1}{{\dfrac{2}{3}}}$
Now reciprocating the denominator we get
$ \Rightarrow S = \dfrac{3}{2}$
Hence the sum of infinite geometric series is found.

Note: Sum of the first $n$ terms of an arithmetic series
$ \Rightarrow {S_n} = \dfrac{{n\left( {{a_1} + {a_n}} \right)}}{2}$
Sum of the first $n$ terms of a geometric series
$ \Rightarrow {S_n} = \dfrac{{{a_1}\left( {1 - {r^n}} \right)}}{{1 - r}}$ $r \ne 1$
Sum of an infinite geometric series with$ - 1 < r < 1$
$ \Rightarrow {S_n} = \dfrac{{{a_1}}}{{1 - r}}$ $r \ne 1$
The sum of the terms in a sequence is called a series.
A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
The sum of the terms in an arithmetic sequence is called an arithmetic series.
The sum of the first $n$ terms of an arithmetic series can be found using a formula.
The sum of the terms in a geometric sequence is called a geometric series.
The sum of the first $n$ terms of a geometric series can be found using a formula.
The sum of an infinite series exists if the series is geometric with $ - 1 < r < 1$ .
If the sum of an infinite series exists, it can be found using a formula.
An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.