Engaging students: Infinite geometric series

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brendan Gunnoe. His topic, from Precalculus: infinite geometric series.

Curriculum:

Students can use the formula for an infinite geometric series to discover the formula for a finite geometric series. The teacher would start by posing the question “Can we use the infinite geometric series to come up with a formula for the finite version?” and writing out a series like so

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

Next, the instructor could ask questions like “If we’re looking for the sum up to the n^th term, where do we need to chop off the terms to get what we want?,” “Does the ending part look familiar?”, and “How can we rewrite the chopped off part so that it looks like what we already know?”. The teacher guides the students into manipulating the formula to get this result

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + ar^{n+1} + \dots$

$\displaystyle \sum_{i=0}^\infty ar^i = ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n + \sum_{j=n+1}^\infty ar^j$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{j=n+1}^\infty ar^j$

The teacher notes that the last sum can be simplified to make it easier to see by doing a substitution of $k = j -n-1$ . Adjusting the bounds and substituting in the new index, we get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1+k}$

$= \displaystyle \sum_{i=0}^\infty ar^i - \sum_{k=0}^\infty ar^{n+1}r^k$

$= \displaystyle \sum_{i=0}^\infty ar^i - r^{n+1} \sum_{k=0}^\infty ar^k$

Note that the two sums are identical, besides the index name, so we can factor and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \sum_{i=0}^\infty r^i$

Lastly, we utilize our formula for an infinite geometric series and get

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = a(1-r^{n+1}) \displaystyle \frac{1}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{1-r^{n+1}}{1-r}$

$ar^0 + ar^1 + ar^2 + \dots + ar^{n-1} + ar^n = \displaystyle a\frac{r^{n+1}-1}{r-1}$

Although the infinite series requires $|r|<1$ , the finite version works for all real $r$ . Although the formal proof that this is the correct formula might be beyond the scope of the intended class, it can easily be done with induction.

Technology:

https://www.khanacademy.org/math/geometry-home/geometry-volume-surface-area/koch-snowflake/v/area-of-koch-snowflake-part-1-advanced

Sal Khan, one of recent history’s most well-known STEM educators, has a fantastic video that shows the relationship between a fractal known as the Koch snowflake and the geometric series. Khan works through the derivation of the formulas for the perimeter and area of an the n^th iteration of the Koch snowflake. It turns out that both the area and perimeters for each iteration can be expressed using a geometric series, but the perimeter diverges to infinity while the area converges. Such a result makes sense intuitively since you can fit every iteration inside of a finite box that is slightly larger than the snowflake, and thus bounding the area, yet it would require an infinitely long wire to go around the perimeter of the limiting shape. Since fractals are not normally included in the math curriculum, showing how math can be used in interesting and different ways to solve problem can be very engaging for students.

Culture:

There is a strong connection between geometric series, fractals, and self-similarity, all with a relatively simple nature. Fractals have been used in architecture and art for a very long time. Examples of self-similarity seen in ancient cultures include Hindu temples, with their structure being composed of self-similar units, and Islamic geometric art found in the domes of mosques.

Since the invention of the computer in the mid-20^th century, more detailed and intricate digital art has been made popular. Although not exactly a geometric series, the Mandelbrot set acts very much like a fractal and was among the first of the uses of a computer to investigate the properties of fractals. It has been used in many ways to make animations, photos and other digital arts.

Another link between fractals and art can be found in the Legend of Zelda games. One of the iconic symbols of the game is called the triforce, which is an equilateral triangle that’s been cut into 4 smaller triangles with the middle piece removed. Such a shape is the first iteration of a fractal known as the Sierpinski triangle. As you can see, fractals can be found in all kinds of art, coming in many different forms.

https://en.wikipedia.org/wiki/Fractal_art

The IRS Uses Geometric Series?

I recently read the delightful article “The IRS Uses Geometric Series?” by Michelle Ghrist in the August/September 2019 issue of MAA FOCUS. The article concerns a church raffle for a $4000 ATV in which the church would pay for the tax bill of the winner. This turned out to be an unexpected real-world application of an infinite geometric series. A few key quotes: According to the IRS rules at the time,

…winnings below a certain level [were] subject to a 25% regular gambling withholding tax…

My initial thought was that the church would need to pay $0.25 \times \$4000 = \$1000$ to the IRS. However, I then wondered if this extra $\$1000$ payment would then be considered part of the prize and therefore also subject to 25% withholding, requiring the church to give $0.25 \times \$1000 = \$250$ more to the IRS. But then this $\$250$ would also be part of the prize and subject to withholding, with this process continuing forever.

I got quite excited about the possibility of an infinite geometric series being necessary to implement IRS tax code. By my calculations… [gave] an effective tax rate of 33-1/3%.

I then read more of the instructions, which clarified if the payer pays the withholding tax rate for the payee, “the withholding is 33.33% of the FMV [Fair Market Value] of the noncash payment minus the amount of the wager.” It was satisfying to discover the behind-the-scenes math leading to that number…

In any event, I am glad to know that the IRS can properly apply geometric series.”

Here’s a link to the whole article: http://digitaleditions.walsworthprintgroup.com/publication/?m=7656&l=1#{%22issue_id%22:606088,%22page%22:%2214%22}

Note: The authors notes that, in January 2018, the IRS dropped the two above rates to 24% and 31.58%.

Engaging students: Finite geometric series

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Caroline Wick. Her topic, from Precalculus: finite geometric series.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Finite geometric series was a concept that began over 4500 year ago in ancient Egypt. The Egyptians used this method of finite geometric series mainly to “solve problems dealing with areas of fields and volumes of granaries” but used it for many other uses too, including the pyramids and math problems similar to those one might find on a STAAR test today (see D1, and F1).

There are seven houses; in each house, there are 7 cats; each cat kills seven mice; each mouse has eaten 7 grains of barley; each grain would have produced 7 hekat. What is the sum of all the enumerated things?

Years passed and finite geometric series were not revisited until around 350 BC by the Greeks, namely Archimedes, who came up with a solution to the math problem V=1/3Ah by finite calculations instead of limits. In addition, the idea that a finite sum could be procured from an infinite series was created in what is called the “Achilles Paradox” (D2, F2).

Years after this came Mathematicians in the middle ages, like Richard Swineshead or Nicole Oresme, who aided the world by further refining these series. This eventually led to the renowned Physicist Isaac Newton to “discover the geometric series” after studying mathematician John Wallis’s method of “finding area under a hyperbola” (F1). We can attribute almost all of what we know about geometric series’ to these fine gentlemen above, and they can only attribute what they know from the ancient Egyptians and Greeks.

How has this topic appeared in pop culture?

In 2002, PBS came out with a kids’ TV show called CyberChase, which is an entertaining cartoon about a bunch of kids who get pulled into “Cyber Space” to fight the bad guy, named Hacker, all while discovering and using different mathematical concepts that they learned along the way. Eleven seasons have passed since the shows beginning and it is still going strong, but one episode that still sticks out to me was their version of explaining geometric series to kids. The episode was called “Double trouble” and was the 9th episode of the second season. The specific geometric series involved in the episode was doubling, but the “real world” clip at the end stood out more vividly to me. After losing a chess game, the main character has to decide between paying the winner $5.00 or paying one penny for the first space on a chess board, then two pennies on the second, then four on the third, and continuing to double the previous number for every space on the entire chess board. Since the main character thought pennies were less, he decided on the second option, only realize after that he would have to pay way more than $5.00 in the end. This helped me understand the most basic geometric series when I was a kid, and has stuck with me to this day, so I am certain that it has and can stick in other students’ brains as well.

Here is the clip from the show:

How can this topic be used in your students’ future courses in math or science?

The idea of finite geometric series is typically lightly introduced around students’ sophomore year of high school when they take geometry, but it is not really expanded upon/explained until students reach Pre-Calculus. The specific TEKS related to this topic are located under Pre-Calculus in (5), (A)-(E) (Source B1). The concept is brought up again in Math Models with Applications and is used for understanding interest on a balance over a period of time, or “loan amortization.” The ideas can also be used to help understand difference equations that involve heat and cooling over a period of time, and how to predict what the temperature might be in the future, which is a concept that is important in the realm of science too.

When students get to college, finite geometric series are expanded upon even more when they take Calculus classes, and they will learn how to prove a series is finite using induction when they get to their Discrete Classes and Real Analysis classes. In the business realm, they will have to use it to predict monetary sums regarding interest and possible growth in a company, so likely no matter where a student ends up, s/he will have to use this important mathematical concept everywhere.

References:
D1, F1: http://mste.illinois.edu/courses/ci499sp01/students/ambucher/math306geo.pdf
D2, F2: https://www.britannica.com/topic/Achilles-paradox
B1: http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.html

My Favorite One-Liners: Part 35

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Every once in a while, I’ll discuss something in class which is at least tangentially related to an unsolved problems in mathematics. For example, when discussing infinite series, I’ll ask my students to debate whether or not this series converges:

$1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \dots$

Of course, this one converges since it’s an infinite geometric series. Then we’ll move on to an infinite series that is not geometric:

$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots$ ,

where the denominators are all perfect squares. I’ll have my students guess about whether or not this one converges. It turns out that it does, and the answer is exactly what my students should expect the answer to be, $\pi^2/6$ .

Then I tell my students, that was a joke (usually to relieved laughter).

Next, I’ll put up the series

$1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \dots$ ,

where the denominators are all perfect cubes. I’ll have my students guess about whether or not this one converges. Usually someone will see that this one has to converge since the previous one converged and the terms of this one are pairwise smaller than the previous series — an intuitive use of the Dominated Convergence Test. Then, I’ll ask, what does this converge to?

The answer is, nobody knows. It can be calculated to very high precision with modern computers, of course, but it’s unknown whether there’s a simple expression for this sum.

So, concluding the story whenever I present an unsolved problem, I’ll tell my students,

If you figure out the answer, call me, and call me collect.

My Favorite One-Liners: Part 20

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Perhaps the world’s most famous infinite series is

$S = \displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$

as this is the subject of Zeno’s paradox. When I teach infinite series in class, I often engage the students by reminding students about Zeno’s paradox and then show them this clip from the 1994 movie I.Q.

This clip is almost always a big hit with my students. After showing this clip, I’ll conclude, “When I was single, this was part of my repertoire of pick-up lines… but it never worked.”

Even after showing this clip, some students resist the idea that an infinite series can have a finite answer. For such students, I use a physical demonstration: I walk half-way across the classroom, then a quarter, and so on… until I walk head-first into a walk at full walking speed. The resulting loud thud usually confirms for students that an infinite sum can indeed have a finite answer.

For further reading, see my series on arithmetic and geometric series.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 3

At the time of this writing, it is unknown if there are infinitely many twin primes, which are prime numbers that differ by 2 (like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc.) However, significant progress has been made in recent years. However, it is known (Gamma, page 30) the sum of the reciprocals of the twin primes converges:

$\displaystyle \left( \frac{1}{3} + \frac{1}{5} \right) + \left( \frac{1}{5} + \frac{1}{7} \right) + \left( \frac{1}{11} + \frac{1}{13} \right) + \left( \frac{1}{17} + \frac{1}{19} \right) = 1.9021605824\dots$ .

This constant is known as Brun’s constant (see also Mathworld). In the process of computing this number, the infamous 1994 Pentium bug was found.

Although this sum is finite, it’s still unknown if there are infinitely many twin primes since it’s possible for an infinite sum to converge (like a geometric series).

When I researching for my series of posts on conditional convergence, especially examples related to the constant $\gamma$ , the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

Another poorly written word problem (Part 9)

Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy.

While the ball is on the 20-yard line, a defensive end is suddenly cursed so that he commits a penalty every down that causes the following:

a. The ball is moved half the distance to the goal line, and
b. The down is replayed.

Show that the ball will eventually travel the entire 20-yard distance to the goal.

Sigh. The textbook expects students to use the formula for an infinite geometric series

$\displaystyle \sum_{n=0}^\infty ar^n = \displaystyle \frac{a}{1-r}$

with $a = 10$ and $r = 0.5$ . However, this series only works if there are an infinite number of terms, so that any finite partial sum will be less than 20. Therefore, saying that the ball “will eventually travel” all 20 yards is misleading, as this implies that this happens after a finite amount of time.

Useless Numerology for 2016: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post.

Part 1: Introduction.

Part 2: $2016 = 2^{10}+2^9+2^8+2^7+2^6+2^5 = 2^{11} - 2^5$ .

Part 3: $2016 = 1 + 2 + \dots + 63$ .

Part 4: $2016 = (1 + 2 + \dots + 9)^2 - (1+2)^2 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3$

Part 5: $2016 = \displaystyle \sum_{n=0}^{63} (-1)^{n+1} n^2$

Zeno’s Hairodox

Source: http://joegp.com/halfcut/

How I Impressed My Wife: Part 4f

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Originally, my wife had asked me to compute this integral by hand because Mathematica 4 and Mathematica 8 gave different answers. At the time, I eventually obtained the solution by multiplying the top and bottom of the integrand by $\sec^2 x$ and then employing the substitution $u = \tan x$ (after using trig identities to adjust the limits of integration).

But this wasn’t the only method I tried. Indeed, I tried two or three different methods before deciding they were too messy and trying something different. So, for the rest of this series, I’d like to explore different ways that the above integral can be computed.

Previously in this series, I have used two different techniques to show that

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x} = \displaystyle \frac{2\pi}{|b|}.$

Here’s my progress so far:

$Q = \displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

$= \displaystyle \int_0^{2\pi} \frac{2 \, dx}{1+\cos 2x + 2 a \sin 2x + (a^2 + b^2)(1-\cos 2x)}$

$= 2 \displaystyle \int_0^{2\pi} \frac{d\theta}{(1+a^2+b^2) + 2 a \sin \theta + (1 - a^2 - b^2) \cos \theta}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\theta}{S + R \cos (\theta - \alpha)}$

$= 2 \displaystyle \int_{0}^{2\pi} \frac{d\phi}{S + R \cos \phi}$

$= \displaystyle -\frac{4i}{R} \oint_C \frac{dz}{z^2 + 2\frac{S}{R}z + 1}$ ,

where this last integral is taken over the complex plane on the unit circle, a closed contour oriented counterclockwise. In these formulas, $R = \sqrt{(2a)^2 + (1-a^2-b^2)^2}$ and $S = 1 + a^2 + b^2$ . (Also, $\alpha$ is a certain angle that is now irrelevant at this point in the calculation).

This contour integral looks complicated; however, it’s an amazing fact that integrals over closed contours can be easily evaluated by only looking at the poles of the integrand. In recent posts, I established that there was only one pole inside the contour, and the residue at this pole was equal to $\displaystyle \frac{R}{ 2 \sqrt{S^2-R^2} }$ .

This residue can be used to evaluate the contour integral. Ordinarily, integrals are computed by subtracting the values of the antiderivative at the endpoints. However, there is an alternate way of computing a contour integral using residues. It turns out that the value of the contour integral is $2\pi i$ times the sum of the residues within the contour; see Wikipedia and Mathworld for more information.