Question

How do you know if an infinite geometric series converges $?$

Answer 1

To know if an infinite Geometric Series converges:

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 $a_1+a_{1}r+a_1{r}^2+a_1{r}^3+...$, where $a_1$ is the first term and $r$ is the common ratio.

An infinite geometric series converges if the absolute value of common ratio of geometric series is less than $1$ that is $\left|r\right|<1$ where $r$ is the common ratio of geometric series.

For example: $\sum _{n=1}^{\infty }10{\left(\frac{1}{2}\right)}^{n-1}$ 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 $12$. Since $\left|\frac{1}{2}\right|<1$ , the sum exits.

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

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

$\begin{array}{rcl}S& =& \frac{10}{1-{\displaystyle \frac{1}{2}}}\\ & =& \frac{10}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\\ & =& 20\end{array}$

Hence, if $-1<r<1$ then the infinite geometric series converges.