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  "body": "\n## Definition\n\nLet $(a_n)_{n \\ge 1}$ be a [[sequence|0QE8STGJH2]] of [[complex numbers|09FTAET5Z4]].\nThe infinite product with factors $a_n$ is the sequence $(p_N)_{N \\ge 1}$ of partial products\n$$p_N = a_1 a_2 \\cdots a_N = \\prod_{n=1}^{N} a_n.$$\nIt is written $\\prod_{n=1}^{\\infty} a_n$.\n\n## Remarks\n\nThe symbol $\\prod_{n=1}^{\\infty} a_n$ also denotes the value of the product when the product [[converges|PCYRWA7SYB]].\nExample: 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.\nSo $p_N \\to \\tfrac12$ and the product converges with value $\\tfrac12$.\nNon-example of convergence: for $a_n = 1 - 1/(n+1)$ the partial products are $p_N = 1/(N+1) \\to 0$.\nBy 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.\nA product can start at another index, such as $\\prod_{n=2}^{\\infty} (1 - 1/n^2)$.\n\n## Explanations\n\n### L1 human\n\nAn infinite product is the list of products of the first $N$ factors, for $N = 1, 2, 3, \\ldots$.\n",
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