Infinite product

Remarks

The symbol $\prod_{n=1}^{\infty} a_n$ also denotes the value of the product when the product converges. Example: for $a_n = 1 - 1/(n+1)^2$ the partial products are $p_N = \prod_{n=2}^{N+1} \frac{(n-1)(n+1)}{n^2} = \frac{N+2}{2(N+1)}$, by telescoping. So $p_N \to \tfrac12$ and the product converges with value $\tfrac12$. Non-example of convergence: for $a_n = 1 - 1/(n+1)$ the partial products are $p_N = 1/(N+1) \to 0$. By convention this product does not converge but diverges to $0$, since a limit $0$ reached by nonzero factors is excluded in the definition of convergence. A product can start at another index, such as $\prod_{n=2}^{\infty} (1 - 1/n^2)$.