It would be very annoying, and more importantly unnecessary, to instead write: ![]() Theorem D: some other theorem whose hypotheses imply those of the ratio test. This proof of Theorem C may seem absurdly indirect: why wouldn't we just cite Theorem A? Well, consider: However, once we have a valid proof of Theorem B, it certainly applies to geometric series: Undoubtedly this is what your notes are trying to say. Of course, the proof of Theorem A cannot use Theorem B, otherwise we have a circular argument. Proof: show that this is implied by theorem A as in the answer by Xander Henderson. Theorem B: Ratio test with usual hypotheses. Its hypotheses do not exclude geometric series, therefore it applies, and its proof must support this. The ratio test is not inadmissible for geometric series. So, when facts about convergence of geometric series in the proof of the ratio test, this sometimes leads to people mistakenly thinking that any time the topic of convergence of geometric series comes up, one must suspend their use of the ratio test and argue from more basic tools. A fair amount of mathematics education involves validating the tools one is already familiar with, which requires temporarily suspending the use of those tools so that we can see how they can be built from more basic tools - and sometimes people learn the wrong lesson and think that's something you always have to do. To elaborate on that last point, people often get stuck in the "tool building" mindset. Possibly due to the author failing to actually convey his meaning.You are reading notes about how to validate the ratio test. ![]() You would have to first derive the facts about geometric series in some other fashion.Īs for your notes, the most likely explanations are: However, if you happen to be in the process of validating the ratio test, it would not be valid to use the ratio test to justify any of the facts you need - such as when geometric series converge - in its validation. It may even be the preferable tool for deciding when a geometric series converges, simply to cut down on the number of things one needs to remember. The ratio test is an example of such a mathematical tool, and is perfectly applicable to geometric series.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |