I’ve posted on infinite series a few times before. But I was inspired to touch on the topic again because I saw this post, yesterday, over at the Math Less Traveled. Actually, the post isn’t really about infinite series as much as it is about p-adic numbers and zero divisors. I’m excited to read more from Brent on this subject. But I digress.
The point I want to make with this post is that students struggle with wrapping their minds around convergent infinite series, and yet they live with them all the time. Students have inconsistently held beliefs about infinite sums.
The simplest convergent series is a geometric series which converges to . The easy proof of this fact goes like this: we look at the sum formula for a finite geometric series, and we notice that
But this proof isn’t very satisfying for the student encountering infinite series for the first time ever. Evaluating the limit feels like ‘magic.’ The idea of adding up an infinite amount of things and getting a finite value is unsettling. I admit, it sounds like quite a lot to swallow. That being said, however, students have no problem declaring the infinite series
to be . It’s not “close to” , it’s not “approaching” , it IS EQUAL TO . And my Precalculus students already accept this as fact. So without even thinking about it, they’ve been living with convergent infinite series all along. Hah!
Once they finally shake their denial, they can more easily accept the convergence of other infinite series like . At first when students encounter a series like this, they think, “surely we can’t say the sum is EQUAL to . It must be close to or approach it, but equal to?” But the same students make no such distinction with .
So there it is. An inconsistently held belief about infinite sums. To the student: You cannot have it both ways. Either you must agree with, or deny, both of the following equations:
But to believe one equation is true and the other is only ‘kind of’ true is inconsistent. I rest my case. 🙂