Consider the following list of numbers: \(4, 8, 16, 32...\) Can you figure out the pattern? How about the sum? What if the list was to go on and on, how would you find the sum if the numbers weren't given to you? In this article, you will look at how to find the sum of infinite geometric series.
Evaluating Infinite Geometric Series
Before you can evaluate an infinite geometric series, it helps to know what one is! In order to do that it can be helpful to break it down and first understand what a sequence is.
A sequence is a list of numbers that follow a specific rule or pattern. Each number in a sequence is known as a term.
There are lots of different types of sequences, including arithmetic and geometric. When thinking about infinite geometric series, it is important to understand what is meant by the term geometric.
A geometric sequence is a type of sequence that increases or decreases by a constant multiple. This is known as the common ratio, \(r\).
Let's look at some examples!
Some examples of geometric sequences include:
- \(2, 8, 32, 128, 512, \dots\) Here the rule is to multiply by \(4\). Notice that the '\(\dots\)' at the end mean the sequence just keeps following the same pattern forever.
- \(6, 12, 24, 48, 96\) Here the rule is to multiply by \(2\).
- \(80, 40, 20, 10, 5\) Here the rule is to multiply by \(\frac{1}{2}\).
Now that you understand what us meant by a sequence, you can think about a series.
A series is the sum of the terms of a sequence.
Let's take a look at some examples.
Some examples of series include:
- \(3+7+11+15+ \dots\) where the original sequence is \(3, 7, 11, 15, \dots\). Again, the '\(\dots\)' means the sum goes on forever, just like the sequence.
- \(6+12+24+48\) where the original sequence is \(6, 12, 24, 48\).
- \(70+65+60+55\) where the original sequence is \(70, 65, 60, 55\).
Now you can consider each of these definitions to fully understand what an infinite geometric series is.
An infinite geometric series is a series that adds up an infinite geometric sequence.
Here are some examples.
Let's go back to the geometric sequence \(2, 8, 32, 128, 512, \dots\). Find the corresponding geometric series.
Answer:
First, you can tell this is a geometric sequence because the common ratio here is \(r = 4\), which means that if you divide any two consecutive terms you always get \(4\).
You could certainly write down that the geometric series is just adding up all of the terms of the sequence, or
\[ 2 + 8 + 32 + 128 + 512+ \dots\]
You could also recognize that there is a pattern here. Each term of the sequence is the previous term multiplied by \(4\). In other words:
\[ \begin{align} 8 &= 2\cdot 4 \\ 32 &= 8 \cdot 4 = 2 \cdot 4^2 \\ 128 &= 32 \cdot 4 = 2 \cdot 4^3 \\ \vdots \end{align}\]
That means you could also write the series as
\[ 2+ 2\cdot 4 + 2\cdot 4^2 + 2\cdot 4^3 + 2 \cdot 4^4 + \dots \]
Remember that the common ratio for this series was \(4\), so seeing a multiplication by \(4\) each time makes sense!
Infinite geometric series have many real-life applications. Take the population for example. Since the population is rising by a percentage each year, studies can be made to predict how big the population will be in \(5\), \(10\), or even \(50\) years to come by using infinite geometric series.
Formula for an Infinite Geometric Series
As you saw in the last example, there is a general formula that a geometric series will follow. The general form looks like:
\[a +a r+ ar^2+a r^3+\dots\]
where the first term of the sequence is \(a\) and \(r\) is the common ratio.
Since all geometric series will follow this formula, take time to understand what it means. Let's look at an example of a series in this form.
Take the geometric sequence \(6, 12, 24, 48, 96, \dots\) . Find the first term and the common ratio, then write it as a series.
Answer:
The first term is just the first number in the sequence, so \(a = 6\).
You can find the common ratio by dividing any two consecutive terms of the sequence. For example
\[ \frac{48}{24} = 2\]
and
\[\frac{24}{2} = 2.\]
It doesn't matter which two consecutive terms you divide, you should always get the same ratio. If you don't then it wasn't a geometric sequence to start with! So for this sequence, \(r = 2\).
Then using the formula for the geometric series,
\[a +a r+ ar^2+a r^3+\dots = 6 + 6\cdot 2 + 6 \cdot 2^2 + 6 \cdot 2^3 + \dots\]
This formula can help you to understand exactly what is happening to each term in order to give you the next term.
Common Ratio of Infinite Geometric Series
You now now how to find the common ratio for a geometric sequence or series, but other than writing down a formula, what is it good for?
- The common ratio \(r\) is used to find the next term in a sequence and can have an effect on how the terms increase or decrease.
- If \(-1<r<1\), the sum of the series can be calculated as an actual number. In this case the series is called convergent.
- If \(r > 1\) or \(r < -1\), the sum of the series won't be a real number. In this case the series is called divergent.
Sum of Infinite Geometric Series
Before we go on to the sum of an infinite geometric series, it helps to remember what the sum of a finite geometric series is. Recall that if you call your series \( a, ar, ar^2, ar^3 , \dots, ar^{n-1} \) then the sum of this finite geometric series is
\[ \begin{align} S_n &= \frac{a(1-r^n)}{1-r} \\ &= \sum\limits_{i=0}^{n-1} ar^i. \end{align}\]
When you have the infinite geometric series \( a, ar, ar^2, ar^3 , \dots \), then the sum is
\[\begin{align} S &= \sum\limits_{i=0}^\infty ar^i \\ &= a\frac{1}{1-r}.\end{align} \]
But remember that the only time \(S\) is a number is when \(-1<r<1\)!
Examples of Infinite Geometric Series
Let's have a look at some examples where you have to identify whether the formula is appropriate and how to use the formula for the sum of infinite geometric series.
If possible, find the sum of the infinite geometric series that corresponds to the sequence \(32, 16, 8, 4, 2, \dots \).
Answer:
To start with it is important to identify the common ratio as this tells you whether or not the sum of the infinite series can be calculated. If you divide any two consecutive terms like
\[ \frac{16}{32} = \frac{1}{2},\]
you always get the same number, so \(r = \frac{1}{2}\). Since \(-1<r<1\) you know that you can actually find the sum of the series.
The first term of the series is \(32\), so \(a = 32\). That means
\[ \begin{align} S &= a\frac{1}{1-r} \\ &= 32\frac{1}{1-\frac{1}{2}} \\ &= 32 \frac{1}{\frac{1}{2}} \\ &= 32\cdot 2 = 64. \end{align}\]
Let's take a look at another example.
If possible, find the sum of the infinite geometric series that corresponds to the sequence \(3 , 6 , 12 , 24 , 48, \dots\).
Answer:
Once again you need to begin with identifying the common ratio. Dividing any two consecutive terms gives you \(r = 2\). Since \(r > 1\) it is not possible to calculate the sum of this infinite geometric series. This series would be called divergent.
Let's look at one more.
If possible, find the sum of the infinite geometric series,
\[\sum^\infty_{n=0}10(0.2)^n.\]
Answer:
This one is already in the summation form! Just like before the first thing to do is find the common ratio. Here you can see that the common ratio is \(r=0.2\). Therefore you are able to complete the sum. You just need to input the information into the formula:
\[ \begin{align} S &= a\frac{1}{1-r} \\ &= 10\frac{1}{1-0.2} \\ &= 10 \frac{1}{0.8} \\ &= 10(1.25) = 12.5. \end{align}\]
Infinite Geometric Series - Key takeaways
- An infinite geometric series is the sum of an infinite geometric sequence.
- When \(-1<r<1\) you can use the formula \[S=\frac{a_1}{1-r}\]to find the sum of the infinite geometric series.
- An infinite geometric series converges (has a sum) when \(-1<r<1\), and diverges (doesn't have a sum) when \(r< -1\) or \(r>1\).
- In summation notation, an infinite geometric series can be written \[\sum^\infty_{n=0}a r^n.\]
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