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  "body": "\n## Definition\n\nLet $(a_n)_{n \\ge 1}$ be a sequence of complex numbers, and consider the [[infinite product|RZ0Z2BTMT4]] $\\prod_{n=1}^{\\infty} a_n$.\nThe product converges if there is an index $m$ such that:\n\n1. $a_n \\ne 0$ for all $n \\ge m$;\n2. the partial products $q_N = \\prod_{n=m}^{N} a_n$ [[converge|0AZ3R52593]] to a limit $q$ as $N \\to \\infty$;\n3. $q \\ne 0$.\n\nThe value of the product is then $\\prod_{n=1}^{\\infty} a_n = a_1 \\cdots a_{m-1} \\, q$.\nA product that does not converge diverges.\n\n## Remarks\n\nThe value does not depend on the choice of $m$.\nIf the product converges, then $a_n \\to 1$, since $a_n = q_n / q_{n-1}$ for $n \u003e m$.\nExample: $\\prod_{n=2}^{\\infty} (1 - 1/n^2)$ converges with value $\\tfrac12$.\nWith $m = 2$, the partial products are $\\prod_{n=2}^{N} \\frac{(n-1)(n+1)}{n^2} = \\frac{N+1}{2N}$, which tend to $\\tfrac12 \\ne 0$.\nNon-example: $\\prod_{n=2}^{\\infty} (1 - 1/n)$ has partial products $1/N \\to 0$, so it fails clause 3 for every $m$ and diverges.\nThe product $0 \\cdot 2 \\cdot 2 \\cdot 2 \\cdots$ also diverges, because its tail products are growing powers of $2$ with no limit.\n\n## Explanations\n\n### L1 human\n\nA convergent product may contain a few zero factors, but the rest must multiply to a nonzero limit.\n",
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