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.

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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 nth 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.

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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 nth­ 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.

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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-20th 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.


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