Mathematics, help needed!

titaniumx3 · 22 Oct 2007 at 22:15

I would really appreciate some help on this transform theory question. I have been at it all day but can't for the life of me figure out how to prove if the Fourier series converges (absolutely) or how it coincides with the given real series. From my lecture notes there are various theorems on convergence and I'm guessing I need to use Dini's Theorem but it doesn't seem to make any sense with regards to the function in question. Here is the question:

I've obtained the Fourier series but can't seem to progress any further.

Any help would be hugely appreciated!

titaniumx3 · 22 Oct 2007 at 22:47

JonC said:
Not done any maths for a couple of years, but I'm sure there are some who have. Give us the Fourier series, see where we can go from there.

Well this is what I got for the Fourier series:

BTW, this is my third year and yes, it's f'ing hard.

titaniumx3 · 22 Oct 2007 at 23:00

FrostedNipple said:
is it as hard in 1st and 2nd year?

1st is pretty easy so long as you put in a fair amount of study. 2nd get's harder since there is more emphasis on analysis and abstraction (with more proofs to do) but 3rd year just takes the urine, lol. That said, I've just started to sort my study routine out so things might get better.

I'm doing the single honours mathematics at university of birmingham, btw.

titaniumx3 · 23 Oct 2007 at 00:59

Amleto said:
do you know about the exp(ix) = cos(x) + isin(x) identity?

Yeah I tried that but it doesn't seem to go anywhere in regards to the series shown in the question.

titaniumx3 · 23 Oct 2007 at 21:00

^^ Yes you are right, I made a mistake whilst tediously drawing out the Fourier series with my mouse, lol. The denominator of the fraction should be n^2 not n.

Anyway, I totally forgot about the odd/even properties of the sine/cosine

, functions so when you expand exp(i*n*x), the even terms cancel out because of (-1)^n and the sine terms sum to zero, hence you eventually get the real series as required.

As for convergence, I'm still confused because in my notes there seem to be several methods to determine convergence, but I just used the fact that the series for 1/n^2 converges hence the fourier series terms also converge, using the comparision test and the fact that |cos(x)| is bounded by 1.

Thanks to everyone for the help!