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Old 2013-03-20, 09:06   #2
NBtarheel_33's Avatar
Jul 2008
Maryland, USA

5·223 Posts

Nitpick-of-the-day: It's actually the Basel problem (after Euler's hometown of Basel, Switzerland). There are, of course, Bessel functions, and knowing Euler, he may well have been involved with them too, but they play no part in this problem. Anyway...

Take a look at this paper/talk on the same subject.

It discusses a lot of the complex analysis behind this problem, and specifically, states that the issue in your step (5) is resolved via the Weierstrass Factorization Theorem, which is a stronger result that comes from the Fundamental Theorem of Algebra.

I also looked at Wikipedia's article on the Basel problem, and it explains that your step (8) is legal because of Newton's identities. The latter article does not resemble anything that I have seen in an undergraduate curriculum, so you might want to just make mention of the buzzword "Newton's identities" in explaining why step (8) is possible.

It's worth mentioning to the "kids" that the upshot is that, as with many problems in the earliest days of calculus and the study of the infinite and infinitesimal, mathematicians were often able to discover these beautiful and unexpected results, but were unable to put them on any sort of rigorous foundation for at least another 100+ years when analysis would be formally established. Nonetheless, having results such as this one would also prove useful as a check on the new rigorous methods that were being implemented.

Another interesting point to mention: Why doesn't this work for the odd powers? What happens if we try to use the cosine series instead of the sine series? Zeta(3) is irrational [cf. Apery, 1978]...but that's about all we know...

Obviously, this result also gives us an approximation for pi (which was part of a talk that I gave at a student session at MathFest in 2004): the quantity sqrt(6 * sum(1/n^2)) should approximate pi for large enough n. How large? How fast is the convergence? (Hint: it's not too bad!)

You might also recommend that they check out William Dunham's books Journey through Genius and Euler: Master of Us All. Both of these books provide nice sketches of Euler's work on this problem (he didn't stop with Zeta(2); rather, he calculated Zeta(n) for even n as high as 18, long before pocket calculators), as well as discussing how he played fast and loose with infinite series.

Good luck! This is a really neat topic and a fine introduction to Euler's amazing and prodigious mathematics.
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