A "Sweet" introduction to Infinite Series

Submitted by Rebecca Rubenstein

Objective: To introduce the main ideas and vocabulary of infinite series

Materials Needed: Cardboard box

Enough donuts and donut holes for demonstration as well as for the whole class

Napkins

A sharp knife

A small cutting board

Graphing calculators

I. Place donuts in box and ask for 3 volunteers. Remove a donut from the box and cut into 4 equal pieces. Give each volunteer 1/4 of donut. Have another student record on board:

Student A Student B Student C Amount remaining

Step 1: 1/4 1/4 1/4 1/4

Cut the remaining 1/4 donut into 4 equal pieces record as

Step 2: 1/16 1/16 1/16 1/16

Continue cutting the remainder into 4 pieces until too small to continue and analyze this process.

Conclude that each student’s share if approaching 1/3.

1/4+1/16+1/64+… Students should be familiar with this convergent geometric series

Repeat this process cutting the donut into 5 pieces. This series gets close to 1/4 of the donut for each student.

Define "pieces" as terms of the series, amounts after each step are partial sums and total amount each student gets is the limit of the series (or partial sums).

II New volunteer who gets 1/2 + 1/3 + 1/4 + 1/5 + 1/6 of donut and ask how many donuts student will have if you keep handing out pieces of 1/n size. Give students some time to think about their answer. Hopefully, someone will realize that if you continue to hand out these pieces the partial sums continue to increase.

Define this series as the harmonic series and say that it diverges since the partial sums do not approach any finite limit.

III Next volunteer. Give this student a whole donut, but before he can take a bite, take back 1/2,then give him 1/3, but take back 1/4. This can be represented by 1-1/2+1/3-1/4+1/5-... and is called the alternating harmonic series. Students will readily agree that the partial sums for this series are less than 1 but greater than 1/2.

(A 60g donut contains approx 3x10²³,so in the first demo about 39 stesps)

With the harmonic series demo, ask how many donuts we’d have after 100 pieces, 1,000, 1,000,000, etc)