Question

How to find the sum of the infinite geometric series \[\dfrac{1}{2} + 1 + 2 + 4 + ...\]?

Answer 1

Hint:We have to find the sum of the given infinite geometric series \[\dfrac{1}{2} + 1 + 2 + 4 + ...\]. For this we will first calculate the absolute value of the common ratio i.e., \[\left| r \right|\]. If \[\left| r \right| < 1\], then the sum will be \[{S_\infty } = \dfrac{a}{{1 - r}}\], where \[a\] is the first term, \[r\] is the common ratio and \[r \ne 1\]. But, if \[\left| r \right| > 1\] then it will be a case of sum to infinity which does not exist.


Complete step by step answer:
In this question, we have to find the sum of the given infinite geometric series \[\dfrac{1}{2} + 1 + 2 + 4 + ...\].
As we know, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.
In general, a geometric series is written as \[a + ar + a{r^2} + a{r^3} + ...\], where the coefficient of each term is \[a\] and the common ratio between the adjacent terms is \[r\].
For the given series we get \[r = \dfrac{1}{{\dfrac{1}{2}}}\] or \[r = 2\].
If \[\left| r \right| > 1\], the terms of the series become larger and larger in magnitude. So, the series does not converge as the sum also gets larger and larger.
In this case, \[\left| r \right| = 2\] which is greater than \[1\].
Therefore, the sum of the infinite geometric series \[\dfrac{1}{2} + 1 + 2 + 4 + ...\] does not exist.

Note:If \[\left| r \right| = 1\] then the series does not converge. The sum of these types of series will depend upon the value of \[r\]. When \[r = 1\], this is an infinite series and all of the terms of the series are the same. But, when \[r = - 1\] then the sum of the terms will oscillate between two values.