Geometric Series

In the fifthcentury BC the Greek philosopher Zeno of Elea posed his famous paradoxes of motion, which were later published by Aristotle. One such paradox stated, "That which is in locomotion must arrive at the half-way stage before it arrives at the goal." The implication of this is that, no matter how far one travels along a straight line towards one’s destination, half of the remaining distance remains yet to be traveled. Before one can travel a distance L, one must travel a distance of L/2, then a distance L/4, and so on. See Figure 1.

Since this subdivision of distances can continue ad infinitum, there are infinitely many tasks to accomplish in traveling from one point to another. Zeno argued that because infinitely many tasks cannot be accomplished in a finite amount of time, motion is impossible. In fact, this apparent paradox has a solution that involves geometric series.

A geometric series is a sum of infinitely many terms such that successive terms have a common ratio r. Symbolically this is

can be expressed more compactly using summation notation as follows.

An example of a geometric series is

where

and

r

= 2/3. Even though a geometric series has infinitely many terms, it turns out that under certain conditions it has a finite sum.

Finite Geometric Series

A finite geometric series is a geometric series that has been truncated to a finite number of terms. Symbolically this is

Since

has finitely many terms, it has a finite sum

. The subscript

reflects the fact that the sum depends on

. Our immediate objective is to derive a formula for

. Consider the quantity

.

Subtracting

from

yields the following.

Solving

for

yields the desired formula.

For example the finite geometric series

, with

,

, and

, has the following sum.

The usefulness of sum formula (8) becomes immediately apparent when one tries to find the above sum by adding all of the terms.

One important application of finite geometric series is in finance. An ordinary annuity is a fund that pays an amount

at the end of each year for

years with annual interest rate

. The future value

of such a fund is given as follows.

The sum in

is a finite geometric series with

and

. So for example if an annuity pays $,2500 at the end of each year for 6 years with an annual interest rate of 8%, the future value of the annuity is

Geometric Series of Constants

A geometric series of constants is precisely what was meant by geometric series in the introductory paragraph of this article. It is given by either

or

, which are presented below as one equation.

Under what conditions does

have a finite sum? If

does have a sum, what is it?

Each of these questions can be answered. The sum

of a geometric series is defined as the value that

approaches as the number

of terms grows without bound. This translates into the notation of calculus as

So the question of whether

has a finite sum is reduced to the question of whether

is a finite

limit

. Looking back at

, the following observations can be made:

If

,

as

, and so

is a finite limit.

If

, the denominator of

is

, and so the limit

does not exist.

If

,

has no limit as

, and so the limit

does not exist.

If

,

does not approach a finite value as

, and so

is not a finite limit.

The above discussion answers the first question that was posed earlier:

has a finite sum as long as

, and it does not have a finite sum otherwise. If the series

has a finite sum

then the series is said to converge to

, and the series is said to diverge otherwise. The question of what that finite sum

is will be addressed now.

As discussed above, if

then

, and so

, so

becomes

So for example the geometric series

converges, since

. Furthermore, it converges to the sum

.

Returning to Zeno’s paradox, in order for a person to travel a straight line distance

, he must first travel half the distance

L

/2, then half of the remaining distance

L

/4, and so on. Expressed in the notation of geometric series, the sum of these infinitely many distances is the following.

Now Zeno’s contention is that it would take an infinite amount of time to travel across the infinitely many, ever-shrinking lengths between the starting and stopping points. And Zeno would be absolutely correct if the infinite series above were to diverge, but it does not. It is a geometric series with

and

, which means that it converges. Care must be taken when finding the sum because the series starts at

. Subtracting off the

term,

can be used as follows.

And so the series yields the eminently sensible result that the sum of all of the partial distances that comprise the journey is equal to the total distance

.

Creating Power Series

A power series is an infinite series of terms of the form

where

is called the center of the power series. Power series are tremendously important in the study of differential equations, among other areas. Consider the rational function

. This

function

is reminiscent of the sum formula

with

and

. It seems natural therefore to suppose that

and this supposition is true, with limits. The series on the right side of

is called a geometric power series for the function

, and it does in fact equal

provided that

. For all

such that

, the power series diverges. Upon comparison with

, it can be seen that the series in

is centered at 0. The series converges on the interval

, which is called the interval of convergence (IOC) of the series. The radius of this interval is 1, which is called the radius of convergence (ROC) of the series. See Figure 2.

Geometric power series representations can be found for many rational functions. For example, a geometric power series representation centered at 1 of the function

is found below.

In this form,

looks like the sum of a geometric series with

and

. The desired representation for

is

converges if and only if

, hence it can be shown that the IOC is

and the ROC is 3.

Bibliography

Anton, Howard, Irl Bivens, and Stephen Davis. Calculus. Hoboken, NJ: Wiley, 2012. Print.

Huggett, Nick. "Zeno's Paradoxes." The Stanford Encyclopedia of Philosophy. Winter 2010. Ed. Edward N. Zalt. Web. 9 Feb. 2015. <URL = http://plato.stanford.edu/archives/win2010/entries/paradox-zeno/>.

Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014. Print.

Ross, Kenneth A. Elementary Analysis: The Theory of Calculus. New York: Springer, 2013. Print.

Stewart, James. Calculus: Early Transcendentals. Belmont, Cal: Cengage, 2012. Print.

Zill, Dennis G. A First Course in Differential Equations with Modeling Applications. Belmont, CA: Cengage, 2009. Print.