I've studied the proof that Euler gave for the famous Basel Problem, and it would seem that while it is technically correct, he does not justify all of his steps properly. Namely, he assumes that
$$\frac{\sin(x)}{x}=\left(1-\frac{x}{\pi}\right)\left(1+\frac{x}{\pi}\right)\left(1-\frac{x}{2\pi}\right)\left(1+\frac{x}{2\pi}\right)\dots$$
simply because they have the same roots, which is really not a strong enough condition. How do you really show that the equality holds?
He then notices that if you use the above equality, and consider it against the Taylor expansion for $\frac{\sin(x)}{x}$, then you can equate the coefficients of the two infinite expansions at each order, and the result of the Basel problem follows. But how do you know that if you have two different expansions for a function, then their coefficients at each order must be equal?
I would really appreciate if someone could show me how to make these two intuitive, yet informal, steps rigorous.
Best Answer
The following theorem can be used: http://en.wikipedia.org/wiki/Weierstrass_factorization_theorem