How can I write $1/n^2$ as a product of n terms $c_1,....,c_n$ in such a way that $c_n$ tends to a limit $\neq 0$?

Question

How can I write $\frac{1}{n^2}$,for each $n\in \mathbb{N}$ with $n\geq 1$, as a product of n terms $c_1,....,c_n$ in such a way that $\lim_{n\rightarrow \infty}c_n\neq 0$?

Answer 1

Put $c_n=1$ and $c_j=n^{\frac{-2}{n-1}}$ for $j=1,2,\dots, n-1$.

Answer 2

$c_n=\frac{c_1c_2\cdots c_n}{c_1c_2\cdots c_{n-1}}=\frac{\frac{1}{n^2}}{\frac{1}{(n-1)^2}}=\frac{(n-1)^2}{n^2}$, with $c_1=1$.

Answer 3

$\dfrac1{n^2} =\dfrac1{n}\prod_{k=1}^{n-1}\dfrac{n-k}{n-(k-1)} $