4 ms·
See the wiki link in a previous comment: https://en.wikipedia.org/wiki/Infinite_product https://en.wikipedia.org/wiki/Infinite_product "The product is said to
by scterg 3y ago
See the wiki link in a previous comment: https://en.wikipedia.org/wiki/Infinite_product https://en.wikipedia.org/wiki/Infinite_product
"The product is said to converge when the limit exists and is not zero."
#1. Consider the series 1 - 1/2^(1/2) + 1/3^(1/2) - 1/4^(1/2) + ... . The sum converges, but the infinite product diverges to zero.
#2. Consider the series x_1 - y_1 + x_2 - y_2 + x_3 - y_3 + ... where x_n and y_n satisfy x_n - y_n = 1/n, x_n * y_n = 1/n. It is obvious that the sum is the harmonic series and diverges. However, the product converges to one, seeing as x_n \to 0 and (1 + x_n) * (1 - y_n) = 1.
- NotYourLawyer 3y agoI like it. I didn’t realize that “diverging to zero” was a thing—wonder if that’s what the author had in mind, or if there’s an example with real divergence.
- scterg 3y agoIndeed. I didn't realize "diverging to zero" with a thing either. I do think this is what the author had in mind, since he writes about the sum of the logs, "whose convergence is equivalent to the convergence of the product".
- NotYourLawyer 3y agoYeah, but that’s in the context of the theorem that requires the terms to be greater than zero.