Last summer, Math With Bad Drawings had a nice series on the notion of infinity that I recommend highly. This topic is a perennial struggle for math majors to grasp, and I like the approach that the author uses to sell this difficult notion.
Part 3 on infinite series and products that are conditionally convergent discusses a head-scratching fact: according to the Riemann series theorem, the commutative and associative laws do not apply to conditionally convergent series.
An infinite series converges conditionally if it converges to a finite number but diverges. Indeed, by suitably rearranging the terms, the sum can be changed so that the (rearranged) series converges to any finite value. Even worse, the terms can be rearranged so that the sum converges to either or . (Of course, this can’t happen for finite sums, and rearrangements of an absolutely convergent series do not change the value of the sum.)
I really like Math With Bad Drawing’s treatment of the subject, as it starts with an infinite product for :
The top line is correct. However, the bottom line has to be incorrect since but each factor on the right-hand side is less than 1. The error, of course, stems from conditional convergence (the terms in the top product cannot be rearranged).
Conditional convergence is typically taught but glossed over in Calculus II since these rearrangements are such a head-scratching topic. I really like the above example because the flaw in the logic is made evidence after only three steps.
In tomorrow’s post, I’ll continue with another example of rearranging the terms in a conditionally convergent series.