There is a simple convergence test for infinite products that I think deserves to be better known.
Theorem. Let be a sequence of positive numbers. Then the infinite product
converges if and only if the series
Proof: Taking the logarithm of the product gives the series
whose convergence is equivalent to the convergence of the product. But observe that
If we assume that , this gives us that
and the theorem follows by the limit comparison test. Q.E.D.
Using this theorem, everything you know about infinite series translates directly to the world of infinite products. For example, the product
converges if and only if .
Before I learned this theorem, I had imagined that there must be an entire theory of convergence for infinite products, as complex and interesting as the theory of series from calculus, but completely unknown to me. Instead, it turns out that no one ever talks about the convergence of infinite products because there is basically nothing new to say!
The Harmonic Series
Another reason I like this theorem is that it gives a nice proof that the harmonic series diverges. According to the theorem, the behavior of the harmonic series is the same as the behavior of the following product:
But this is just
This clearly diverges, for the partial products are the sequence of positive integers.
Finally, here’s a fun little pair of exercises:
1. Find a sequence of real numbers such that converges but diverges.
2. Find a sequence of real numbers such that diverges but converges (and is greater than zero).