Talalwarsi5897 20 Posted October 14, 2014 Report Share Posted October 14, 2014 (edited) Hey, so i thought about exploring the Basel Problem and how Euler solved it after 100 years. It basically asks for the precise summation of the reciprocals of the squares of the natural numbers to infinity, i.e, 1+(1/2^2)+(1/3^2)+(1/4^2)+... So would it be appropriate to investigate the methods and concepts used by Euler and how Riemann managed to build on it years later to define the basic properties of the series using his developed Riemann Zeta function? What i'm trying to ask is whether this would be appropriate or not. Thanks for any feedback. Edited October 14, 2014 by Talalwarsi5897 Reply Link to post Share on other sites

ctrls 184 Posted October 14, 2014 Report Share Posted October 14, 2014 Yes you can, the aim of an IA could be to solve a known problem as seen with this sample IA. I'm not really sure how you'll link it Riemann's work, though I suppose you could include a superficial note on how that leads to his hypothesis. Proving this with SL knowledge is pretty difficult though, even the HL calculus option teaches barely enough content to allow for a rough proof. The first proof posted on wikipedia for example not only assumes some HL knowledge, but certain things aren't fully justified. I know of another proof that which has a bit more justification in the working, but it still requires the knowledge of more advanced ideas like the binomial series and integration by reduction formulae - neither of which are on the HL maths syllabus. One thing to be careful with these kinds of explorations is that you don't end up copying a proof posted on the internet - make sure you fully understand what's going on and explain the proof as clearly as possible in your own words. Lastly, you'll probably want to mention ideas relating to convergence and constantly ask yourself if it's valid to manipulate infinite series in certain ways, as many results about finite sums don't extend to infinite ones. It's inevitable that you'll make manipulations that aren't fully justified with these sums, so you probably want to have some discussion/consideration of it. 1 Reply Link to post Share on other sites

Talalwarsi5897 20 Posted October 19, 2014 Author Report Share Posted October 19, 2014 Yes you can, the aim of an IA could be to solve a known problem as seen with this sample IA. I'm not really sure how you'll link it Riemann's work, though I suppose you could include a superficial note on how that leads to his hypothesis. Proving this with SL knowledge is pretty difficult though, even the HL calculus option teaches barely enough content to allow for a rough proof. The first proof posted on wikipedia for example not only assumes some HL knowledge, but certain things aren't fully justified. I know of another proof that which has a bit more justification in the working, but it still requires the knowledge of more advanced ideas like the binomial series and integration by reduction formulae - neither of which are on the HL maths syllabus. One thing to be careful with these kinds of explorations is that you don't end up copying a proof posted on the internet - make sure you fully understand what's going on and explain the proof as clearly as possible in your own words. Lastly, you'll probably want to mention ideas relating to convergence and constantly ask yourself if it's valid to manipulate infinite series in certain ways, as many results about finite sums don't extend to infinite ones. It's inevitable that you'll make manipulations that aren't fully justified with these sums, so you probably want to have some discussion/consideration of it.Thanks so much, i will keep that in mind.Do you think that it is a fair enough topic or would you advise me to change it? Reply Link to post Share on other sites

ctrls 184 Posted October 19, 2014 Report Share Posted October 19, 2014 Thanks so much, i will keep that in mind. Do you think that it is a fair enough topic or would you advise me to change it? It really depends on how much you know about summing infinite series and whether you can actually understand the solution. If you find yourself struggling and not understanding what's going on however, then I would advise you to change. You are awarded a maximum of 6 marks for the use of mathematics, you don't get extra marks for tackling an exceptionally difficult topic. In all honestly though, I don't think this topic is suitable for an SL student. Even most HL students would likely find this very difficult. Reply Link to post Share on other sites

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