The Basel Problem: Why is 𝜋 here?

What is the sum of the reciprocal of the squares:

Also known as the Basel Problem, this sum converges to a finite value (approximately ~1.6449). This can be calculated using an infinite product and Taylor series expansion of the sine function.

To write the sine function as an infinite product, we will examine its zeros. As seen in its sinusoidal shape, sine has zeros at 0, 𝜋, -𝜋, 2𝜋, -2𝜋, etc. This can be written as the product:

To find a Taylor polynomial for the sine function centered at zero, we will calculate terms from taking successive derivatives.

These two definitions of sine will be equal to each other. We will then expand both sides to find the coefficient of the x² term.

Note: Ox^n is used to define the arbitrary higher-order terms.

Since the two polynomials are equal, the x² terms must also be equal.

And we have solved the Basel Problem. The fact that 𝜋 appears here (and is squared) is not intuitive, but the math speaks for itself!