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Geometric Series: Convergence and Divergence

CYNTHIA HELZNER, Christopher Haines, Kathryn Boddie
  • Author
    CYNTHIA HELZNER

    Cynthia Helzner has tutored middle school through college-level math and science for over 20 years. She has a B.S. in microbiology from The Schreyer Honors College at Penn State and a J.D. from the Dickinson School of Law. She also taught math and test prep classes and volunteered as a MathCounts assistant coach.

  • Instructor
    Christopher Haines
  • Expert Contributor
    Kathryn Boddie

    Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.

Convergence and divergence of a series in math follows some specific rules. Learn the rules as well as the geometric series convergence test. Also see examples. Updated: 03/15/2022

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).


The amount of distance left until the finish line, point F, is equal to the last distance traveled, which is half of the prior distance traveled.

The runner runs from point A to points B, C, D, then E.  The distances traveled are 4, 2, 1, and one half.  The finish line is point F.


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

Speed on each sub-interval

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 vn is the distance traveled after n seconds.

Distance traveled after n seconds

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

Distance simplification

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

Increment for geometric series

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.

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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.

The Theorem

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

Partial sums defined

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.

Limit of partial sum

Definition of infinite series

Geometric series

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, ....

Geometric Sum Formula finite n

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:

case i

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

Geometric series formula

Case ii

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

Case ii result sum value

case iii divergent oscillation

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).

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Video Transcript

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

Speed on each sub-interval

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 vn is the distance traveled after n seconds.

Distance traveled after n seconds

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

Distance simplification

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

Increment for geometric series

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:

Partial sums defined

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.

Limit of partial sum

Definition of infinite series

Geometric series

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, ....

Geometric Sum Formula finite n

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:

case i

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

Geometric series formula

Case ii

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

Case ii result sum value

case iii divergent oscillation

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).

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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

converges to
if |r|<1. In general, if "a" is a constant, then a geometric series of the form
converges to
if |r|<1. Determine if the following series converge or diverge - and if they converge, find the value the series converges to.

Solutions

1. Since we have

the geometric series converges. It converges to

2. Since we have

the geometric series converges. It converges to

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

Then, since
the geometric series diverges.

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

Then, since
the geometric series converges. It converges to

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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