Compute $\lim_{x\to0} \frac{\cosh x\cosh 2x\cosh 3x \cdots \cosh nx-1}{x^2}$ Compute:
$$\lim_{x\to0} \frac{\cosh x\cosh 2x\cosh 3x \cdots \cosh nx-1}{x^2}$$
How would you tackle this problem? Thanks.

Note that
$$
\cosh \alpha=1+\frac{\alpha^2}{2}+o(\alpha^2)
$$
for $\alpha\to0$. So for $x\to 0$  we have
$$
\prod\limits_{k=1}^n \cosh kx= 
\prod\limits_{k=1}^n \left(1+\frac{k^2 x^2}{2}+o(k^2 x^2)\right)=
\prod\limits_{k=1}^n \left(1+\frac{k^2 x^2}{2}+o(x^2)\right)=
1+\sum\limits_{k=1}^n\frac{1}{2}k^2 x^2+o(x^2)
$$
Hence
$$
\lim\limits_{x\to 0}\frac{\prod\limits_{k=1}^n\cosh kx - 1}{x^2}=
\lim\limits_{x\to 0}\frac{\frac{1}{2}\sum\limits_{k=1}^n k^2 x^2+o(x^2)}{x^2}=
\frac{1}{2}\sum\limits_{k=1}^n k^2=\frac{n(n+1)(2n+1)}{12}
$$
