Question

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

Answer 1

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.