A "sweet" Introduction to Infinite Series

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Materials
  • Credits

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    Objective: Introduce the main ideas and vocabulary of infinite series 
                            and the convergence of series.

    Level:  Calculus II or Calculus III.

    Prerequisites: Understanding of sequences and convergence of sequences.

    Platform: A classroom and a 50-minute period.

    Instructor's Notes:
    This demonstration came about several years ago when I was trying to think of ways to help students sort out the concepts and vocabulary of infinite series, and to understand the fundamental idea of convergence of a series. 

    As mathematics teachers, a "lets imagine that .. " scenario and formal definitions probably worked just fine for us when we were students, but lots of my students are in engineering and other applied fields, and many work better physically and visually than they do abstractly. I wanted something that would get their attention and would accurately introduce the abstract ideas of the terms of a series, the partial sums of a series, and convergence or divergence of a series. In fact, in one 50 minute lecture/demonstration/feast we get all that plus alternating series and the divergence of the Harmonic series.

    Materials: 

    • cardboard box
    • a dozen donuts and enough donut holes for the whole class
    • napkins
    • a large knife
    • a small cutting board


    Approximate script: (A narration of using the donuts with the class.)

    I come into class with everything inside the closed box. The fact that I’m bringing a box to class automatically gets more attention than usual.

    I ask for 2 volunteers and pick them from near the front of the class. Then I hold up the large knife, and ask for a third volunteer. That usually gets more attention, and I pick the third volunteer from near the front again. Finally, I give each volunteer a napkin and I take out the cutting board and put a donut on it.

    I tell the class that I am going to divide up a donut among the volunteers in a strange mathematical way, and we are going to keep track of how much each volunteer gets. (I also have them get out their calculators.)

    Step 1: I cut the donut into 4 (equal) pieces and give each of the three volunteers
                 a piece. Then I go to the board and record 
     
                   student A           student B             student C         amount left
    Step 1:         1/4                      1/4                        1/4                      1/4 
    Step2: Then I cut the remaining 1/4 donut into 4 equal pieces and give each 
                 student one piece. Now the recording is
                   student A           student B             student C         amount left
    Step 1:         1/4                      1/4                        1/4                      1/4 
    Step 2:         1/16                    1/16                      1/16                   1/16

    so each student has a total of 1/4 + 1/16 = 5/16.
     

    Step 3: Then I cut the remaining 1/16 donut into 4 equal pieces and give each 
                 student one piece.

                 With this action on the board I add the line

                Step 3:          1/64                 1/64                      1/64                   1/64

                so each student has 1/4 + 1/16 + 1/64 = 21/64 .

    After one or two more steps the pieces are getting rather small so I simply eat the last small piece and suggest that we analyze what happens if I had continued the process. 


    The class is very willing to agree that the "amount left" is quickly approaching zero, so in the "limiting case," the whole donut is distributed equally among the three students and each student gets 1/3 of the donut:

    .

    At this point I usually repeat the process with 4 different students as volunteers and cut the donut (and each remainder) into 5 pieces. This gives us a series that gets closer and closer to 1/4 of the donut for each student:

    .

    Now I go back to the board and start attaching names to the different parts of the information we have recorded:

    • the "pieces" 1/4, 1/16, 1/64, ... are called the "terms" of the series
    • the "amounts after each step", 1/4, 5/16, 21/64, .. are called the           "partial sums" of the series (abbreviated ps for use below)
    • and since the total amount each student is getting in our first donut sharing example was approaching 1/3, we say that "the limit of the series" is 1/3      (or "the limit of the partial sums" is 1/3).
    I usually ask for predictions about the following series:

    and mention that we will show tomorrow that the predictions are correct.
     

    Onward

    Next I ask for 1 volunteer and pick someone in the middle or near the back of the room. This person gets 

      • 1/2 of a donut (partial sum = 1/2), 
      • then 1/3 (ps = 5/6), 
      • then 1/4 (ps= 13/12 = 1.083333), 
      • then 1/5 (ps = 77/60 = 1.283333 ), 
      • then 1/6 (ps = 87/60 = 1.45 ).
    About now I stop and ask for suggestions about how many donuts the student will acquire if I keep handing out pieces of size 1/n

    After recording several of these student estimates on the board, I suggest that perhaps a calculator isn't the answer to everything and that some brain power will do very nicely.

    Writing

    I start by grouping the terms in the "usual" way

    and justify that the sum in each group is greater that or equal to 1/2 (so after 8 pieces we have a total greater that 3/2.).

    Next I include the terms 

    so now the partial sum is greater than 4/2= 2.

    Now I include the terms

    so the partial sum is greater than 5/2.

    I mention that if we keep going, the partial sum will eventually be larger than 3 donuts, 10 donuts, a billion donuts, etc. and so we can declare the volunteer to be the "king or queen of the universe of donuts"

    I wrap things up here by mentioning that this series, 

    is called the Harmonic series, and that we say it diverges since the partial sums do not approach any finite limit. (Later in the course when some student mistakenly concludes that since the terms of some series approach zero then the series must converge, I refer them to the king or queen of donuts.)

    The rewards of mathematics -- sugar for everyone

    At this point I usually put the donut holes and the napkins in a couple spots in the room and invite everyone to partake of the holes; they have 2 minutes to get a couple and go back to their seats.

    The give and take of donut series

    Finally, I usually have time for one last example. I get out a large fancy frosted donut and ask for one last volunteer. 

    I give the student the whole donut (which pleases them), but before they can take a bite, I take back 1/2 of the donut (ps = 1/2). Then I give them 1/3 but take back 1/4 (ps = 7/12). Usually I go to the board and ask how to represent what has been happening as a series and hopefully someone suggests

    which I tell them is called the Alternating Harmonic series.

    Using the usual arguments, the students are willing to accept that the partial sums of the Alternating Harmonic series are less than 1 and greater than 1/2, and then between 1/2 and 5/6, and then between 7/12 and 5/6, and usually someone with a calculator will tell me that the partial sums are close to 0.62??? .

    Class Time is up

    By now we are always out of time, and I just mention that at the next class we will formalize some of this reasoning and start to examine some additional series.

    Later classes

    In future classes when the vocabulary is defined and used, I refer back to the "donut meaning" of the words, and to the results for the donut (geometric series) and the donut Harmonic series.

    A few additional comments

    A colleague in chemistry calculated that a "typical" 60g donut (see the 'Beginning' picture above) contains approximately 3 x 1023 atoms. So in the first demo (cutting each piece into fourths), after about 39 steps we are subdividing a single atom.

    With the Harmonic series, I sometimes point out that even though we know that the "king or queen of the donut universe" eventually has an unlimited number of donuts, it takes a lot of pieces to acquire even a few entire donuts:

      • after 100 pieces they only have a bit more than 5 entire donuts           (about 5.187)
      • after 1,000,000 pieces they still have less than 15 entire donuts             (about 14.392)
      • and it takes more than 1043 pieces to get to 100 entire donuts.
    Certainly we could do this whole process as an "imagine that ..." situation, but actually doing the physical cutting and distributing (and adding) makes it much more real and memorable and enjoyable for the students. (Many of my foreign students seem particularly surprised that a mathematics teacher would actually give them sweets in class.)

    Bagels or apples or other foods work, but in morning classes the donuts are very popular (and I have a sweet tooth).

    For under $10 I definitely have their attention while I introduce the main vocabulary and some early results of infinite series. (Later, when they ask when the next donut lecture is, I mention that my salary only affords one per term but that they are free to contribute to my sweet tooth.)

    The student comments at the end of the term always include a few positive comments about the "donut lecture", and many students seem to appreciate that I tried to make series real and physical and tasty for them. The demo is very lo–tech, but I always have fun on donut day.

    Credits:  This demo was submitted by 

    Dale Hoffman
    Department of Mathematics 
    Bellevue Community College

    and is included in Demos with Positive Impact with his permission.



    DRH 2/14/00   Last updated 5/19/2006

    Since 3/1/2002