Infinite Geometric Series

An infinite geometric series is the sum of an infinite geometric sequence . This series would have no last term. The general form of the infinite geometric series is $a_1+a_{1}r+a_1{r}^2+a_1{r}^3+\mathrm{...}$ , where $a_1$ is the first term and $r$ is the common ratio.

We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you won't get a final answer. The only possible answer would be infinity. So, we don't deal with the common ratio greater than one for an infinite geometric series.

If the common ratio $r$ lies between $-1$ to $1$ , we can have the sum of an infinite geometric series. That is, the sum exits for $\left|r\right|<1$ .

The sum $S$ of an infinite geometric series with $-1<r<1$ is given by the formula,

$S=\frac{a_1}{1-r}$

An infinite series that has a sum is called a convergent series and the sum $S_n$ is called the partial sum of the series.

You can use sigma notation to represent an infinite series.

For example, ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}10{\left(\frac{1}{2}\right)}^{n-1}}$ is an infinite series. The infinity symbol that placed above the sigma notation indicates that the series is infinite.

To find the sum of the above infinite geometric series, first check if the sum exists by using the value of $r$ .

Here the value of $r$ is $\frac{1}{2}$ . Since $\left|\frac{1}{2}\right|<1$ , the sum exits.

Now use the formula for the sum of an infinite geometric series.

$S=\frac{a_1}{1-r}$

Substitute $10$ for $a_1$ and $\frac{1}{2}$ for $r$ .

$S=\frac{10}{1-\frac{1}{2}}$

Simplify.

$\begin{array}{l}S=\frac{10}{\left(\frac{1}{2}\right)}\\ =20\end{array}$