# Geometric Series: Convergence and Divergence

## Convergence and Divergence

A **series** is the sum of a **sequence**, which is a list of numbers that follows a pattern. An **infinite series** is the sum of an infinite number of terms in a sequence, such as 2 + 4 + 6 + 8 + ... Infinite series are subject to convergence and divergence.

In Zeno's Paradox of the Dichotomy, a runner must always reach the halfway point between her current position and the finish line. Those distances will become half as large each time. Thus, there will always be some distance left to go, and so she will never reach her destination. This is similar to a car that reduces its speed by half each second so that it will travel half of the prior second's distance. Like the runner, the car will never reach its destination. In both of these cases, the distances traveled become smaller by half each time, and so the series of distances traveled converges to a set limit (the total distance of the trip).

In contrast, saving one penny on Day 1 and then doubling the amount saved each day would result in infinite savings. Thus, the series of money saved each day does *not* converge to a set limit, and so it diverges.

### What Does Converge Mean?

The first question to understand is: what does "converge" mean? **Converge** means to approach a set limit, such as 2 or -350.25. In the context of geometric series, the absolute values of the terms will get smaller, and so, as with the runner, there will always be some distance left between the limit and the current sum of the terms. So, for those wondering, "What is a convergent series; what is a divergent series?" A convergent series is simply a series that approaches a set limit, and a divergent series is a series that does not approach a set limit.

### What Does Diverge Mean?

**Diverge** means to fail to approach a set limit. In the context of geometric series with |r| > 1, the absolute values of the terms will get larger, causing the series to approach infinity or negative infinity.

While many divergent series becomes infinitely large, not all divergent series will do so. All that is required for a series to be divergent is for it to fail to converge. For example, the terms of {eq}\displaystyle\sum_{n=1}^{\infty}(-1)^{n} {/eq} alternate between -1 and 1, so the sum alternates between -1 and 0. Therefore, the series diverges since it does not approach a set limit.

## Convergence/Divergence Series

### A Traveler's Paradox

Let's say a traveler is driving a car down a road that is one mile long. At the start, the car is moving 60 miles per hour. After one second, the car immediately reduces its speed to 30 miles per hour. After two seconds, the car instantaneously reduces its speed from 30 miles per hour to 15 miles per hour. More generally if *v(t)* is the speed of the car at time *t*, then

This is where *n* = 1, 2, 3, 4, 5,......

In the formula, the time *t* is in seconds, and the speed *v(t)* is in feet per second, and *v*` n` is the distance traveled after

*n*seconds.

The distance traveled after *n* seconds takes the form of a **geometric sum**. This sum simplifies to

By passing this equation, which we've called (1), to the limit as *n* tends to infinity, we can then see that the car's distance will approach 176 feet as the time *t* tends to infinity. However since the expression given in (1) is strictly less than 176, the car will never reach the end of the road. Instead, the car slows down so drastically that it gravitates towards the 176 foot marker. The marker is seen as a point of attraction along the one mile road. It appears counter-intuitive that even though the car never reduces its speed to zero, it never comes anywhere close to reaching the end of the road - it's not even one-thirtieth of the distance!

Figure 1 illustrates the car's traveling progression. The first car represents the position at zero seconds, the second car at one second, the third at two seconds, and so on (from left to right). As is apparent, the one second increments decrease with time. We can also see this from the identity

The moral of this story is that a sum having an infinite number of terms can possibly end up being finite. We've given an example of a convergent geometric series, making the concept of a convergent series more precise.

## Convergence Theorem

There are a variety of convergence theorems and tests for a variety of series types. In essence, any convergence theorem states that if a series approaches a high or low limit, it converges.

### When Does a Series Converge?

As explained above, a series converges if it approaches a finite limit. So, what does that mean in concrete terms, i.e., when does a series converge? For a geometric series, the series converges if and only if the absolute values of the terms get smaller. For example, some geometric series that converge are:

- {eq}\displaystyle\sum_{n=1}^{\infty}4(2)^{-n} = 2 + 1 + 0.5 + 0.25 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}10(-0.1)^{n} = 1 + -0.1 + 0.01 + -0.001 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}\left (\frac{1}{5}\right)^{n - 1} = 1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} +\space ... {/eq}

Note that, for each convergent series, the terms get closer to zero.

## Convergence Tests

Convergence tests are used to determine whether a series will converge. The particular test to use varies based on the characteristics of the series being analyzed, but, in general, the tests determine whether the sum of the terms approaches a finite limit. Some tests do this by determining whether the absolute values of the terms get smaller, while others use limits and integrals from calculus.

### Geometric Series Convergence Test

The geometric series convergence formula is {eq}\frac{a}{1-r} {/eq} if |*r*| < 1, where *a* is the first term and *r* is the **common ratio**, i.e., the number that each term is multiplied by in order to produce the next term.

Some people refer to it as a formula, but it is both a formula and a test. It is a formula for finding the infinite geometric series, while its requirement that |*r*| < 1 (which may also be written as -1 < *r* < 1) is a test of whether the series converges. This is sometimes referred to as the **Geometric Series Theorem**: a geometric series will converge if -1 < *r* < 1; otherwise it will diverge.

### Will the Series Converge or Diverge?

Use the geometric series convergence formula to solve the following problem. Will the series converge or diverge? If it does converge, what is the infinite sum?

{eq}\displaystyle\sum_{n=1}^{\infty}2(0.9)^{n} {/eq}.

Examine the series to see if |*r*| < 1. Here, *r* = 0.9 so |*r*| < 1. Thus, the series converges.

So, use the geometric convergence formula to see what the infinite sum is. Here, *a* is 1.8 because the term value is 1.8 when *n* = 1 i.e. {eq}2(0.9)^{1} = 1.8 {/eq}.

So, {eq}\frac{a}{1-r} = \frac{1.8}{1-0.9} = 18 {/eq}. Thus, the series converges to 18.

## Convergent Series Examples

Before looking at some convergent series examples, consider what makes a series convergent:

- The sum of the terms approaches (but does not reach) a set limit.
- The absolute values of the terms get smaller.
- The terms get closer to zero.
- For a geometric series, |
*r*| < 1.

### Geometric Series Convergence

A geometric series written in standard form is {eq}\displaystyle\sum_{n=1}^{\infty}ar^{n-1} {/eq} or {eq}\displaystyle\sum_{n=1}^{\infty}ar^{n} {/eq}. For a geometric series written in standard form, the base of the exponent is *r* so, if the base of the exponent is less than 1, then |*r*| < 1 and the series will converge. If the geometric series is not written in standard form, plug-in two consecutive values of *n* and divide the second term by the first term to get *r*.

To understand geometric series convergence, examine the following examples of geometric series that converge:

- {eq}\displaystyle\sum_{n=1}^{\infty}2(0.5)^{n} = 1 + 0.5 + 0.25 + 0.125 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}(-0.1)^{n-1} = 1 + -0.1 + 0.01 + -0.001 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}\left (\frac{3}{2}\right)^{-n} = \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \frac{16}{81} +\space ... {/eq}

Note that, for each convergent series, |*r*| < 1 and the terms get closer to zero regardless of the signs of the individual terms. The first two examples are written in standard form, so *r* is the base of the exponent. The third example is not written in standard form because of the *-n*, so divide two consecutive terms to find *r* as follows: {eq}r = \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{2}{3} {/eq}.

## Divergence Series Examples

All that is required for a series to be divergent is for it to not approach a set limit. Thus, any of the following would make a series divergent:

- The sum of the terms does not approach a set limit.
- The sum of the terms alternates between two set values.
- The sum of the terms approaches infinity or negative infinity.
- The absolute values of the terms get larger.
- The terms get further from zero.
- For a geometric series, {eq}|r| \geq 1 {/eq}.

### Divergent Function

Just as a divergent series is a series that does not approach a set limit, a divergent function is a function that does not approach a finite limit (horizontal asymptote). For example, the terms of {eq}\displaystyle\sum_{n=1}^{\infty}\frac{1}{16}\cdot2^{n-1} {/eq} can be modelled by the function {eq}f(x) = \frac{1}{16}\cdot2^{x-1} {/eq}, as shown in the graph. In this function, the *y*-value doubles each time *x* increases by 1. The terms thus get further from zero and do not level off.

### Geometric Series Divergence

Here are a few examples of geometric series that diverge:

- {eq}\displaystyle\sum_{n=1}^{\infty}2^{n} = 2 + 4 + 8 + 16 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}(1)^{n-1} = 1 + 1 + 1 + 1 +\space ... {/eq}

- {eq}\displaystyle\sum_{n=1}^{\infty}2(-1)^{n} = -2 + 2 + -2 + 2 +\space ... {/eq}

Note that, for each divergent series, {eq}|r| \geq 1 {/eq} and so the terms do not approach zero. In the first example, the terms get further from zero. In the second and third examples, the terms do not get further from zero, but the series still diverge because the terms do not get closer to zero. In the second example, the terms themselves have a finite limit, which is 1, but the *sum* of the terms does not have a finite limit.

## Lesson Summary

A **series** is the sum of a **sequence**, and an **infinite series** is the sum of an infinite sequence. Some infinite series **converge** to a finite limit. Those that do not are said to **diverge**. The **Geometric Series Theorem** gives the values of the **common ratio**, *r*, for which the series converges and diverges: a geometric series will converge if *r* is between -1 and 1; otherwise, it will diverge. If the series converges, then the infinite sum is {eq}\frac{a}{1-r} {/eq}, where *a* is the first term and *r* is the common ratio. To find *a*, plug in the first value of the variable into the term formula. In sigma notation, the term formula is what is written after the sigma, the variable is the letter before the = sign under the sigma, and the first value of the variable is what is written after the = sign under the sigma.

Just as series can converge or diverge, functions can converge or diverge as well. A convergent function is a function that approaches a set limit (i.e., it has a horizontal asymptote). A divergent function is a function that does not approach a set limit (i.e., no horizontal asymptote). Divergent series and functions often approach positive or negative infinity, but they need not do so; any series or function that fails to converge to a finite limit is divergent.

## The Theorem

To begin, we define the theorem, which we'll call equation 2:

where *r* does not equal 0 if the limit of a partial sum exists as a real number. We write the definition of an infinite series, like this one, and say the series, like the one here in equation 3, converges.

If the aforementioned limit fails to exist, the very same series diverges. It's denoted as an infinite sum whether convergent or divergent.

The partial sums in equation 2 are geometric sums, and this is because the underlying terms in the sums form a **geometric sequence**. The respective infinite series is then called a **geometric series**.

Which values of *r* does the series given in (3) converge? It turns out that there is a convenient, shortcut formula for the partial sums defined in (2). The formula can be proven directly by induction, and for that reason, we omit the proof here.

Here's the Lemma: Geometric Sum Formula in equation 4:

For *n* = 0, 1, 2, 3, 4, ....

As an immediate consequence of equation 4, we arrive at the **geometric series theorem**. This theorem gives the value of *r* for which the series converges and diverges.

We first let the partial sums be defined as in (2) and the respective infinite series defined as in (3). Then, we have the following results:

In this case, we get our geometric series formula and case ii:

In this second case, we get the following resulting sum value in case ii and the divergent oscillation in case iii.

In this third and final case, the series cannot be assigned any value, since the limit of the partial sums does not exist even in the extended real number sense. The monotone convergence theorem ensures the existence of the limits in cases i and ii (as extended real numbers).

### A Traveler's Paradox: Revisited

We now generalize the preceding example to the following. Suppose for *n* = 1, 2, 3, 4, 5, .... we use the general form of the velocity function where *r* does not equal 0:

Then by (4), the position after *n* seconds is

By using the fact that the velocity is constant between successive integer seconds, the position at time *t* is seen to be defined by the formula appearing here:

Note that negative *r* allows the car to move in reverse. Then depending on whether the absolute value of *r* is less than one, the vehicle's position will either hone in on a fixed location or exhibit increasingly larger distances from the initial location. You can see how this is all played out with the tables and figures that appear. You don't have to memorize all of the content in these tables and figures; it's just important that you see all of the data in one place:

Here's Table 1a in which *r* = 0.5:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 132 154 165 170.5 173.25 174.625 175.3125 175.65625 175.828125 |
88 44 22 11 5.5 2.75 1.375 0.6875 0.34375 0.171875 |

Here's Figure 2a, in which *r* = 0.5:

Now, here's Table 1b, in which *r* = -0.5:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 44 66 55 60.5 57.75 59.125 58.4375 58.78125 58.609375 |
88 -44 22 -11 5.5 -2.75 1.375 -0.6875 0.34375 -0.171875 |

Now, here's Figure 2b, in which *r* = -0.5:

Now, here's Table 1c, in which *r* = -2:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 -88 264 -440 968 -1848 3784 -7480 15048 -30008 |
88 -176 352 -704 1408 -2816 5632 -11264 22528 -45056 |

Now, here's Figure 2c, in which *r* = -2:

Now you should understand and be able to actually see how both convergence and divergence of a series can play out.

## Lesson Summary

Let's take a couple of moments to review what we've learned. We learned that a **geometric series** is simply a constant ratio between terms that follow one another. We gained our insight of a geometric series by studying a vehicle's movement along a one-dimensional axis. The partial sums form a sequence of real numbers which tends to a real number when convergent. If this does not occur, the geometric series diverges. The series can diverge in two different ways, and this depends on whether *r* is positive or negative.

We also learned that the **geometric series theorem** gives the value of *r* for which the series converges and diverges. The concept of convergence/divergence extends to a broader class of series. The geometric series is one of the few series where we have a formula when convergent that we will see in later sections.

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## Convergence/Divergence Series

### A Traveler's Paradox

Let's say a traveler is driving a car down a road that is one mile long. At the start, the car is moving 60 miles per hour. After one second, the car immediately reduces its speed to 30 miles per hour. After two seconds, the car instantaneously reduces its speed from 30 miles per hour to 15 miles per hour. More generally if *v(t)* is the speed of the car at time *t*, then

This is where *n* = 1, 2, 3, 4, 5,......

In the formula, the time *t* is in seconds, and the speed *v(t)* is in feet per second, and *v*` n` is the distance traveled after

*n*seconds.

The distance traveled after *n* seconds takes the form of a **geometric sum**. This sum simplifies to

By passing this equation, which we've called (1), to the limit as *n* tends to infinity, we can then see that the car's distance will approach 176 feet as the time *t* tends to infinity. However since the expression given in (1) is strictly less than 176, the car will never reach the end of the road. Instead, the car slows down so drastically that it gravitates towards the 176 foot marker. The marker is seen as a point of attraction along the one mile road. It appears counter-intuitive that even though the car never reduces its speed to zero, it never comes anywhere close to reaching the end of the road - it's not even one-thirtieth of the distance!

Figure 1 illustrates the car's traveling progression. The first car represents the position at zero seconds, the second car at one second, the third at two seconds, and so on (from left to right). As is apparent, the one second increments decrease with time. We can also see this from the identity

The moral of this story is that a sum having an infinite number of terms can possibly end up being finite. We've given an example of a convergent geometric series, making the concept of a convergent series more precise.

## The Theorem

To begin, we define the theorem, which we'll call equation 2:

where *r* does not equal 0 if the limit of a partial sum exists as a real number. We write the definition of an infinite series, like this one, and say the series, like the one here in equation 3, converges.

If the aforementioned limit fails to exist, the very same series diverges. It's denoted as an infinite sum whether convergent or divergent.

The partial sums in equation 2 are geometric sums, and this is because the underlying terms in the sums form a **geometric sequence**. The respective infinite series is then called a **geometric series**.

Which values of *r* does the series given in (3) converge? It turns out that there is a convenient, shortcut formula for the partial sums defined in (2). The formula can be proven directly by induction, and for that reason, we omit the proof here.

Here's the Lemma: Geometric Sum Formula in equation 4:

For *n* = 0, 1, 2, 3, 4, ....

As an immediate consequence of equation 4, we arrive at the **geometric series theorem**. This theorem gives the value of *r* for which the series converges and diverges.

We first let the partial sums be defined as in (2) and the respective infinite series defined as in (3). Then, we have the following results:

In this case, we get our geometric series formula and case ii:

In this second case, we get the following resulting sum value in case ii and the divergent oscillation in case iii.

In this third and final case, the series cannot be assigned any value, since the limit of the partial sums does not exist even in the extended real number sense. The monotone convergence theorem ensures the existence of the limits in cases i and ii (as extended real numbers).

### A Traveler's Paradox: Revisited

We now generalize the preceding example to the following. Suppose for *n* = 1, 2, 3, 4, 5, .... we use the general form of the velocity function where *r* does not equal 0:

Then by (4), the position after *n* seconds is

By using the fact that the velocity is constant between successive integer seconds, the position at time *t* is seen to be defined by the formula appearing here:

Note that negative *r* allows the car to move in reverse. Then depending on whether the absolute value of *r* is less than one, the vehicle's position will either hone in on a fixed location or exhibit increasingly larger distances from the initial location. You can see how this is all played out with the tables and figures that appear. You don't have to memorize all of the content in these tables and figures; it's just important that you see all of the data in one place:

Here's Table 1a in which *r* = 0.5:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 132 154 165 170.5 173.25 174.625 175.3125 175.65625 175.828125 |
88 44 22 11 5.5 2.75 1.375 0.6875 0.34375 0.171875 |

Here's Figure 2a, in which *r* = 0.5:

Now, here's Table 1b, in which *r* = -0.5:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 44 66 55 60.5 57.75 59.125 58.4375 58.78125 58.609375 |
88 -44 22 -11 5.5 -2.75 1.375 -0.6875 0.34375 -0.171875 |

Now, here's Figure 2b, in which *r* = -0.5:

Now, here's Table 1c, in which *r* = -2:

Elapsed Time (Seconds) | Distance Traveled (Feet) | Distance Increment (lag one difference) |
---|---|---|

0 1 2 3 4 5 6 7 8 9 10 |
0 88 -88 264 -440 968 -1848 3784 -7480 15048 -30008 |
88 -176 352 -704 1408 -2816 5632 -11264 22528 -45056 |

Now, here's Figure 2c, in which *r* = -2:

Now you should understand and be able to actually see how both convergence and divergence of a series can play out.

## Lesson Summary

Let's take a couple of moments to review what we've learned. We learned that a **geometric series** is simply a constant ratio between terms that follow one another. We gained our insight of a geometric series by studying a vehicle's movement along a one-dimensional axis. The partial sums form a sequence of real numbers which tends to a real number when convergent. If this does not occur, the geometric series diverges. The series can diverge in two different ways, and this depends on whether *r* is positive or negative.

We also learned that the **geometric series theorem** gives the value of *r* for which the series converges and diverges. The concept of convergence/divergence extends to a broader class of series. The geometric series is one of the few series where we have a formula when convergent that we will see in later sections.

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## Additional Examples

In the following examples, students will practice determining whether a Geometric Series converges or diverges and find what the Geometric Series converges to; if it does converge. The examples start as very basic and develop to involving more algebraic manipulation as the student progresses. After completing the examples, students will be better able to recognize a geometric series and how to determine its convergence.

## Examples

Recall that a geometric series of the form

*a*" is a constant, then a geometric series of the form

## Solutions

1. Since we have

2. Since we have

3. We need to rewrite this series so that it is in the proper form. We have

4. We need to rewrite this series so that it is in the proper form. We have

#### What is convergence and divergence series?

A series is the sum of a sequence of numbers. A convergent series is a series that approaches a set limit, while a divergent series is a series that does not approach a set limit.

#### How do you know if its convergence or divergence?

Pick a convergence or divergence test based on the type of series. For example, to test a geometric series, see if |r| < 1. If so, the geometric series converges; if not, it diverges. In contrast, an arithmetic series always diverges.

#### What does a convergent geometric series mean?

A geometric series is the sum of a sequence of numbers in which each term is multiplied by the same number, r, to produce the next term. A convergent geometric series is a geometric series that approaches a set limit.

#### How do you know if a geometric series converges?

If |r| < 1, the absolute values of the terms get smaller, so the geometric series converges. The quantity r is the common ratio, i.e., what each term is multiplied by to produce the next term.

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