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# Comprehensive Guide on Geometric Series

*schedule*May 20, 2023

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# Finite and infinite geometric series

The finite geometric series is defined as follows:

Where:

$a$ is the starting value.

$r$ is known as the common ratio and represents the multiplicative factor in which the terms in the series change.

In contrast, the infinite geometric series has an infinite number of terms:

## Computing finite geometric series by hand

Consider the following finite geometric series:

In this case, the starting value is $a=3$ and the common ratio is $r=2$. Notice how each successive term in the series is multiplied by the common ratio.

An example of an infinite geometric series with the same starting value and common ratio is:

# Formula for the sum of finite geometric series

The sum of a finite geometric series is given by:

Where:

$a$ is the starting value of the geometric series.

$r$ is the common ratio ($r\ne1$).

$n$ is the number of terms in the series.

Proof. Recall that the finite geometric series is defined as:

Now, we multiply both sides by the common ratio $r$ to get:

Let's write $S_n$ and $rS_n$ on top of each other:

We've added a $0+$ term in the beginning of $rS_n$ to align $S_n$ and $rS_n$. Now, we subtract $rS_n$ from $S_n$ to get:

Notice how $S_n$ is undefined when $r=1$. Let's think about the case when $r=1$. When the common ratio is equal to one, the geometric series is:

Even though our formula \eqref{eq:ByzMoH9Wot7R2cVsndo} will not allow us to compute the sum of the series when the common ratio is equal to one, the sum is still defined and is computed by $na$.

## Computing sum of finite geometric series by formula

Consider the following finite geometric series:

Compute the sum of the series.

Solution. The starting value is $a=3$ and the common ratio is $r=2$. Let's use formula for the sum of finite geometric series:

Notice how the formula extremely handy when computing a series with a large $n$.

# Formula for the sum of infinite geometric series

If the common ratio $r$ is between $-1$ and $1$, that is $\vert{r}\vert\lt1$, then the sum of infinite geometric series is given by:

Where:

$a$ is the starting value of the geometric series.

$r$ is the common ratio between $-1$ and $1$.

Note that if $\vert{r}\vert\gt1$, then the infinite geometric series diverges to infinity.

Proof. Given $\vert{r}\vert\lt1$, we know that:

This means that as $n$ becomes larger and larger, $r^n$ approaches zero. For instance:

Again, this is only true when $\vert{r}\vert\lt1$.

Now, recall that the sum of a finite geometric serieslink is given by:

To obtain an infinite geometric series, we let $n$ tend to infinity and use the properties of limits to simplify:

This completes the proof.

## Computing the sum of infinite geometric series by formula

Consider the following infinite geometric series:

Compute the sum of the series.

Solution. The starting value is $a=3$ and the common ratio is $r=1/2$, which is between $-1$ and $1$. Therefore, we can use the formula for the sum of infinite geometric series:

This means that as we consider more and more terms in the series, the sum converges to $6$.