Solved Problem 1 - Generalized Basel Problem for "N" Nested Infinite Sums



RESULT


Proof

Let us start with a Maclaurin series and work our way to the generalized result.

Dividing both sides of the Maclaurin series of sinx by x we get:
Since the zeroes of sinx are integral multiples of π, we can use the Weierstrass factorization theorem to represent the above function as a product involving its zeroes:
Expanding the above product:
We want π in the numerator in the RHS of the final result, so let us factor out the reciprocals of the even powers of π in the above expression:
Rewriting the above expression in the general form:
Equating the above expression to our original expression for sinx/x:
Equating the corresponding coefficients of powers of x in the LHS and RHS above and multiplying both sides by powers of , we get the Basel problem generalized for n nested infinite sums:



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