All entries for Friday 04 May 2012

May 04, 2012

Cauchy Condensation Test

The Cauchy condensation test is a convergence test. It’s also, from what I can see, one of the few we didn’t cover in Analysis. Which is a shame, because it’s rather nice.

For a positive non-increasing sequence f(n), the sum \sum_{n=1}^{\infty}f(n) converges if and only if the sum \sum_{n=1}^{\infty}2^nf(2^n) converges.

A sketch proof in one direction should be rather evident: as the sequence is non-increasing, we can replace every group of length 2^n by its initial value. For the reverse, the idea is similar.

Let us consider our old friend \sum_{n=1}^{\infty}\frac{1}{n^p}. Consider \sum_{n=1}^{\infty}2^n(\frac{1}{2^n})^p=\sum_{n=1}^{\infty}2^{n-np}=\sum_{n=1}^{\infty}2^{n(1-p)} which is a geometric series and convergent if and only if p>1.

A1: Note–Making and Reading but Rather More of the Former

Dr Laura Davies; 3rd May.

Looks like a slight (major :P) adaptation of the Cornell method could be useful; in particular having a summary of every page would be nice. Unfortunately I’ve no longer lectures so this won’t be something I’ll be able to put into practice as such. My current favoured techniques are plain linear on lined paper for lectures, and a sort of “patchwork” or “tortoiseshell” method, which takes a blank sheet of A4, starts writing at the top left and, when a section is completed, draws a line around it and continues writing. It’s rather dense, quick and fun to write, but not too simple to read quickly. I tend to transfer notes made this way to the computer afterwards.

One interesting point brought up was noting the books you read: where you read them, and interesting points. This seems like a good idea - I recently found a message in which I’d commented that the book in question contained a countable covering of R2, but hadn’t noted which book (it was probably Apostol’s Analysis, based on the person I was talking to). This seems a good idea, although I wonder how to do it nicely - it would seem easy to get cluttered.

One point made was having a pen in hand to encourage specificity and focus. While reading from a screen this seems a decent idea, while reading from a book I doubt I’ll do it (my specialty is certainly reading from a screen).

The last of interest were comments (however brief) on skimming/scanning. Scanning seems to be what the Survey part of SQ3R intends, instead of “contemplate the issue”, so that’s something to try. My current outlook on the issue is that of the fellow who has learnt to move his hands at ridiculous speeds and so has never learnt of touch-typing, as his current methods allow him to type perfectly well (to use a somewhat strenuous metaphor) - that is, my default reading speed is rather fast, so I’ve never had to skim to be able to read something within a time limit.

So, let us:
#Note books (probably in a .txt, I like those files)
#Pen in hand, paper ready

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