{
  "aliases": [
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  "body": "\n## Definition\n\nLet $(a_n)$ be a real sequence.\nThe series $\\sum a_n$ converges conditionally if $\\sum a_n$ [[converges|0CVVKY7GY2]] and $\\sum |a_n|$ diverges.\n\n## Remarks\n\nSo a convergent series converges either [[absolutely|0W3K3QPEWT]] or conditionally, and not both.\nThe [[alternating harmonic series|0JQ5Y3WWVE]] converges conditionally.\nBy the [[Riemann rearrangement theorem|0E7VBYGFF7]], the terms of a conditionally convergent series can be reordered to give any real sum.\n\n## Explanations\n\n### L1 human\n\nA conditionally convergent series converges only because its positive and negative terms cancel.\nThe positive terms alone add up to $+\\infty$, and the negative terms alone add up to $-\\infty$.\n",
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  "msc": [
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  "name": "Conditional convergence",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/02Y/02YRQXZDCS.md",
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