learning about manifolds, teaching sequences and series.

i'm giving myself a week to learn about manifolds or rather, riccι curvature. it seems like one of those things that everyone should know but that few actually know well.

it's not that i want to become a geometer,
i just want to give a talk about analysιs on manifolds, that's all ..

.. then again, it would be nice to work on more concrete spaces. if i learn enough about them, then maybe i can prove something about them.

[shrugs]
a boy can dream, right?

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on an unrelated note, i love teaching sequences and series. it took me a while until i figured out why: it's the closest thing to analysis that you can teach in a standard calculus course.

my students may hate the comparιson test, but i quite like it. there's nothing like estimating something when you don't have to compute it. q-:

i know that the stewarτ textbook doesn't cover the root test, because i was tempted to teach it to my students, but decided against it. are there other "standard" textbooks which do cover it?

then again, it could be that i like them too much:

i think my students are ill at ease with series and convergence tests, because every time i show them an example or two where a particular test works, i also show them a non-example where it either cannot be applied or that it gives no conclusion. [1]

you'd think that this wouldn't make much of a difference. students seemingly understand that some definite integrals are better off done with substitution, rather than by parts ..

.. but show them a series, and suddenly they freeze.

[1] i likened the ratio test to a "magic 8-ball" in that sometimes it just tells you: "reply hazy, try again .."