I have been trying to learn and figure out the elementary rigorous proof posted here:
Basel problem - Wikipedia, the free encyclopedia
under "A rigorous proof"
I know there are...simpler proofs, but because of the elementary knowledge needed to understand this, I'm trying to learn this one.
I wasn't sure if this belonged in the calculus section, but it has some elements of calculus in it.
For the proof, I know most of the background knowledge. I understand De Moivre's theorem, the binomial theorem, one-to-one functions, trig identities, the inequality given, limits, and the squeeze theorem.
I'm not completely sure about Viete's formulas...I know that it gives us a value for the sum of roots.
Anyway, I somewhat understand this proof but not completely.
In the first step, I know how the (cotx+i)^n function is derived and how it is expanded.
I see how imaginary and real parts are grouped and the sine function is set equal to the imaginary parts.
In the next step it says 0 is equal to that equation and I understand why its equal to 0. However, I do not see why it ends with (-1)^m. I think I see why it would be a '1' since the 'i' was factored out, but I do not understand why it is negative and why it's raised to the mth power.
In the next equation, a function p(t) is defined. I was wondering why (cotx)² needs to be a one-to-one function in the interval in order for this to happen. I also do not see why the equation before it must equal 0 for this to happen.
The steps following I generally understand the processes.
It would be greatly appreciated if a run through for the proof was posted, but most importantly I would like the clarifications to be clarified.
Thank you very much.