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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence with $a_n \\ge 0$ for all $n$.\nThen the series $\\sum a_n$ converges if and only if its [[partial sums|0NB643K60Q]] $s_N = \\sum_{n=0}^{N} a_n$ are bounded above.\nIn this case, $\\sum_{n=0}^{\\infty} a_n = \\sup_N s_N$.\n\n## Remarks\n\nThe partial sums increase, since $s_{N+1} - s_N = a_{N+1} \\ge 0$. The result is then the [[monotone convergence theorem|0FM6N4R0VB]].\nIf the partial sums are not bounded, they tend to $+\\infty$, and one writes $\\sum a_n = +\\infty$.\nThe [[comparison test|04Z5G0THN6]], the [[integral test|019QZ6RRVH]] and the [[Cauchy condensation test|0B96KMDDD1]] all rest on this fact.\n\n## Explanations\n\n### L1 human\n\nWhen all terms are nonnegative, the running total can only go up.\nSo it either stays below some ceiling and converges, or it grows past every bound.\n",
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