Evaluate using Taylor polynomials: lim as x to 0 (cos x

Evaluate using Taylor polynomials: $\displaystyle\lim_{x\to0}\frac{\cos x\cos2x\cos3x\cdots\cos nx-1}{x^2}$.

Step 1: Expand each factor to order $x^2$

\[ \cos kx = 1-\frac{k^2x^2}2+o(x^2), \qquad k=1,\ldots,n. \tag{1} \]

Step 2: Expand the product

In a product of $n$ factors of the form $1+\varepsilon_k$, the order-$x^2$ contribution comes from choosing $\varepsilon_k$ from exactly one factor at a time (cross terms are $O(x^4)$):

\[ \prod_{k=1}^n\cos kx = 1+\sum_{k=1}^n\left(-\frac{k^2x^2}2\right)+o(x^2). \tag{2} \]

Step 3: Sum of squares

Using $\sum_{k=1}^nk^2=\dfrac{n(n+1)(2n+1)}6$: \[\prod_{k=1}^n\cos kx = 1-\frac{n(n+1)(2n+1)}{12}x^2+o(x^2). \tag{3}\]

Step 4: Verification

\[n=3,\ x=0.001\text{: numeric value}\approx-6.99998,\ \text{matching }{-}\tfrac{3\cdot4\cdot7}{12}=-7\ \checkmark\]