The Everything Seminar

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2021-07-30 14:30:10

Using this theorem, everything you know about infinite series translates directly to the world of infinite products. For example, the product

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:

This entry was posted on January 26, 2008 at 7:18 am and is filed under Basic Grad Student, High School, Jim, Undergraduate. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

A similar related result is used often in probability. The result is that for real numbers with , the infinite product converges to a nonzero real number if and only if the sum converges. The proof is essentially the same as the one given above.