Question

How do you find the sum of the infinite geometric series $1 - x + {x^2} - {x^3} + {x^4} - ...?$

Answer 1

Hint: An infinite geometric series is the series given and, as we know, the common ratio of consecutive terms in a geometric series is fixed. To find the sum of infinite geometric series, firstly, by dividing a term with its progressive term, find the common ratio between terms. And then use the following formula:
${S_\infty } = \dfrac{a}{{1 - r}},\;{\text{where}}\;{S_\infty },\;a\;{\text{and}}\;r$ are sum of infinite geometric series, first term of the series and common ratio of the series respectively.

Formula used:
Common ratio of a G.P.: $r = \dfrac{{{u_{n + 1}}}}{{{u_n}}}$
Infinite sum of G.P.: ${S_\infty } = \dfrac{a}{{1 - r}}$

Complete step by step solution:
In order to find the sum of the infinite geometric series $1 - x + {x^2} - {x^3} + {x^4} - ...$ we will first find the common ratio of the series as following
$r = \dfrac{{{u_{n + 1}}}}{{{u_n}}},\;{\text{where}}\;{u_{n + 1}}\;{\text{and}}\;{u_n}$ are “n+1th” and “nth” term of the geometric series respectively
We will take second and first term, to find the common ratio,
$ \Rightarrow r = \dfrac{{ - x}}{1} = - x$
Now, we will use the formula for sum of infinite terms of geometric series which is given as follows
${S_\infty } = \dfrac{a}{{1 - r}},\;{\text{where}}\;{S_\infty },\;a\;{\text{and}}\;r$ are sum of infinite geometric series, first term of the series and common ratio of the series respectively
In the series $1 - x + {x^2} - {x^3} + {x^4} - ...$ the first term is $a = 1$
Putting $a = 1\;{\text{and}}\,r = - x$ in the above formula, we will get
\[{S_\infty } = \dfrac{1}{{1 - ( - x)}} = \dfrac{1}{{1 + x}}\]
Therefore the required infinite sum of the given series is equals to \[\dfrac{1}{{1 + x}}\]

Note: The infinite sum of the following series will only exist when value of “x” will be less than one and greater than negative one excluding zero, that is $x \in \left( { - 1,\;1} \right)\sim0$, otherwise the infinite sum of the given series will not exist and also if $x = 0$ then series itself will not exist.