On Fri, 16 Nov 2012, math-fun-request@mailman.xmission.com wrote:
If you try to provide a similar epsilon-delta proof that the series ...9999.0 sums to -1, you will fail, and I can provide an epsilon-delta proof that this series does *not* have a limit.
Yes. Looking more closely (by a factor of 1+epsilon) though, the result you really have is that if the epsilon-delta definition is the correct definition of what it means to be a limit, then there is no limit. So either it's not the limit, or the definition is bogus. What we're really exploring here is whether the second option is the case. The interesting historical question is how we came to agree that e-d is the correct definition. There are multiple ways that Euler's intuitive approach to limits could have been formalized, and somehow we settled on this particular one, which fails to capture important parts of his intuition. This is historically interesting because in other cases, like the definition of the definite integral, there is still no consensus but we're mostly OK with that. We are taught the Riemann definition in introductory calc, but eventually we may want to work in more complicated spaces than Rn or handle functions that Riemann fails on where some other definition will win out. -- Tom Duff. Linux? Is that an OS, like Pentium?