Dealing with Infinity
Sunday 2 November 2014
There comes a time in both Maths HL or Maths SL when probing questions about infinity will be confronted - by both teacher and student - or, at least they should be. This usually first occurs with the topic of infinite geometric series. The concept of infinity is a difficult concept - not just for young students, but for anyone (see blog post Is infinity a number?). So, you should not discourage students asking challenging questions about infinity. Often the brightest students are the ones that are the most uncomfortable with accepting that the sum of the following infinite geometric series actually has a finite sum.
This should be perplexing to anyone - and students will be perplexed, if they really think about it. They are familiar with finite processes - e.g. add these two numbers, factorise this polynomial with three terms, etc. An infinite series is an infinite process - and funny things happen with infinite processes.
I've found that it's nice to have some things in my teaching toolbox to address student confusion and discomfort with some infinite geometric series having a finite sum. Using the series shown above with first term of 1/2 and common ratio of 1/2, I will show them the square displayed above left. The visual representation of the series helps some of them accept that the sum is equal to exactly 1 rather than just "really close to 1."
For the students who still don't believe that it's possible that an infinite series could have a finite sum, I tell them to turn on their calculator and enter the number 0.9999999999 ; decimal point followed by ten 9s. Then enter the number 0.9999999999999 ; decimal point followed by thirteen 9s. Ask them to comment on what they observe. [note: The number of 9s which will cause the calculator to display 1 will differ between calculator models due to memory size. For a TI-84 Plus C Silver Edition, this occurs with eleven 9s; and for the TI-Nspire CX it occurs after thirteen 9s - as shown below]
A calculator will state that 0.9999999999999 = 1 because of memory limitations. The number 0.9999999999999 is a rational number which is clearly not equal to 1. It's the sum of the following finite series.
This strange (but understandable) result on their calculator gets students thinking - and then I present the following proof. After this I may have some stubborn students who are still unwilling to accept that an infinite series may have a finite sum - but, in my experience, most students will be more willing to accept it by the time I show the following proof that 0.999999 ... = 1 (now, an infinite number of 9s after the decimal place).