Glad to say I've finished TMA04 for M208. This was the first analysis block and all that reading of Brannan's book paid off. The TMA covered the following topics

1) Proof of various inequalities and finding the greatest lower bound of a function

2) Testing varying sequences for convergence

3) Testing varying series for convergence

4) Testing to see if various functions are continuous at a given point

Have to say I found it fairly straightforward, not that analysis is, but the TMA essentially was a run through of various tests which are relatively straightforward to apply. The main problem, as with M337, is to make sure every detail is included, thus for example when proving a sequence made up of a whole sequence of terms it's not enough to divide by the dominant term and then caclulate the basic limit you have to include such phrases as because

$$ {1/n},{1/n^3} $$ etc are basic null sequences, by the Combination rules it can be seen that the sequence converges to a and so forth. All of which is so easy to forget and I guess if I do such questions in the exam I'll lose a few marks for not including every detail. The only way to guarantee success is to look carefully at the way answers to the examples in the course units have been structured and learn it off almost by rote.

On the other hand even the answers given in the course units miss out essential details, for example they quite often to forget to mention that cos(x) or sin(x) is only bounded if

$$ x \epsilon R$$ where R is the set of real numbers, Or for example a term like

$$c=\sqrt{\frac{a}{b-m'}}$$ which is often encountered when trying to find the least upper bound or greatest lower bound of a sequence is only real if b-m' > 0. Thus an appeal to the Archimedean princple namely that if $$a \epsilon R$$, there is an integer n such that n > a only works if the term c is real. Again a point not stressed by the units.

Coo getting more pedantic than the M208 answers must be learning something

Seriously though all this at first sight pedantry is essential if one wants to be sure of ones conclusion, for example as my mate Neil has pointed out it took mathematicians over 2000 years to resolve Zeno's paradox properly.

As a final point it's a bit worrying that the TMA's only appear to be scratching the surface still I suppose to avoid complaints that the TMA's are to hard I suppose needs must. I can always get a sanity check by looking at the Cambridge Maths examples sheets for analysis.

Anyway a day off from TMA questions tomorrow who hoo.

Hi Chris,

ReplyDeleteThankfully I can remember enough of the analysis blocks from last year to follow what you're writing. :-)

Exploring convergence and divergence was great fun.

I don't think M208 deals with the Euler-Mascheroni constant which makes for interesting reading in the case of the harmonic series where the sum of n terms can be approximated by ln(n) + 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 ...........

There's a bit on it here http://www.jimloy.com/algebra/hseries.htm , if it catches your eye... got to be easier than adding 1 + 1/2 + 1/3 + ... + 1/999999 + 1/1000000 to arrive at 14.384 :-)

The reading list you mention at my blog would be gratefully received by many, including myself, I'm sure.

That constant actually turns up in Integrals which are used in Particle physics

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