The Sine Function: Taylor Series and Infinite Product

The Taylor (or Maclaurin) series for sine is a well-known infinite sum. However, through Euler's brilliant insight into factoring power series, we also have a remarkable infinite product representation.

1. The Taylor Series (Maclaurin Series)

The Taylor series for \(\sin(x)\) about \(x = 0\) is an infinite sum of terms:

\[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\]

2. The Infinite Product Representation

Euler discovered that the sine function can be expressed as an infinite product, which reveals its roots directly:

\[\sin(x) = x \prod_{n=1}^{\infty} \left( 1 - \frac{x^2}{n^2 \pi^2} \right)\]

3. Intuition and Derivation

The key idea comes from complex analysis. The sine function, \(\sin(z)\) with \(z \in \mathbb{C}\), is an entire function. Its zeros are at \(z = n\pi\) for all integers \(n\).

A polynomial with these roots would be \(P(z) = z \prod_{n=1}^{\infty} (z^2 - n^2\pi^2)\), but this product does not converge. The Weierstrass factorization theorem provides the correct, convergent form, which is the infinite product shown above. Each factor \((1 - \frac{z^2}{n^2\pi^2})\) is zero exactly when \(z = \pm n\pi\), matching the roots of the sine function.

4. A Famous Special Case: The Wallis Product

Substituting \(x = \pi/2\) into the infinite product yields the famous Wallis Product for \(\pi/2\):

\[\sin\left(\frac{\pi}{2}\right) = 1 = \frac{\pi}{2} \prod_{n=1}^{\infty} \left( 1 - \frac{1}{4n^2} \right)\]

Rearranging gives:

\[\frac{\pi}{2} = \prod_{n=1}^{\infty} \left( \frac{4n^2}{4n^2 - 1} \right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots\]

Summary

Representation	Formula
Taylor Series (Infinite Sum)	\(\displaystyle \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}\)
Euler's Infinite Product	\(\displaystyle \sin(x) = x \prod_{n=1}^{\infty} \left( 1 - \frac{x^2}{n^2 \pi^2} \right)\)

While the Taylor series expresses \(\sin(x)\) as an infinite sum of polynomial terms, Euler's product expresses it as an infinite product of linear factors that explicitly show its roots. Both are beautiful and profoundly useful in different contexts.