8
$\begingroup$

We introduce infinite sequences and series very thoroughly in calculus classes. We first define infinite sequences, then series, carefully discussing notions of convergence, etc., and discuss all sorts of rules for convergence before allowing students to see Taylor's theorem.

However, suppose that one just went to the board and wrote down $$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots$$ without making a general definition of an infinite series, or explaining anything about convergence. (Presumably one would have to explain factorials so that the pattern is clear.)

Or, more simply, one could write $$1 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$

I have had interesting debates with colleagues as to whether this is a good idea -- and our debates seem to rely on an empirical question.

Are these formulas easily comprehensible to, say, Calc I students, bright high school students taking competitions, or other students who have not had formal exposure to infinite series? Or are infinite series a genuine conceptual stumbling block to students?

For example, would students be able to see how the first formula allows them to quickly find accurate approximations for $e$?

  • 4
    Yeah, when we write $\frac{1}3=0.333...$, we are implicitly talking about infinite series. Although "1=0.9999...." still yields problems for young people who aren't clear on the formal nature of what the sum "actually" means.2012-12-12
  • 1
    The problem is, if you wrote $f(x)=1+x + 2!x^2+...$ your students would have no idea that that never converged. And if you wrote $1+x+x^2+...$ you would have to be specific that this only converges if $|x|<1$.2012-12-12
  • 0
    I would say that students have a little more intuition about infinite sequences and convergence. That said, it can be difficult to transfer this intuition to infinite series.2012-12-12
  • 0
    Not what you're asking, but related: "[Motivating Infinite Series](http://math.stackexchange.com/questions/9524/motivating-infinite-series)"2012-12-12

2 Answers 2