{
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  "body": "\n## Statement\n\nLet $\\sum a_n$ be a real series that [[converges conditionally|02YRQXZDCS]].\nLet $M$ be a real number.\nThen there is a [[rearrangement|0HRJ2Y74T3]] $\\sum a_{\\sigma(n)}$ that converges to $M$.\nThere is also a rearrangement that diverges to $+\\infty$, and one that diverges to $-\\infty$.\n\n## Remarks\n\nThe positive terms and the negative terms each give a divergent series, and $a_n \\to 0$. Take positive terms until the partial sum exceeds $M$, then negative terms until it falls below $M$, and repeat.\nSo a real series converges absolutely if and only if all its rearrangements converge to the same sum.\nFor the [[alternating harmonic series|0JQ5Y3WWVE]], take two positive terms, then one negative term, and repeat. The sum becomes $\\frac{3}{2} \\ln 2$.\n\n## History\n\nAugustin-Louis Cauchy is usually credited with the first remark that a new order can change a sum. He studied the alternating harmonic series in 1833.\nThe theorem is named after Bernhard Riemann.\n\n## Explanations\n\n### L1 human\n\nA conditionally convergent series has positive terms with an infinite total and negative terms with an infinite total.\nBy a careful choice of order, we can steer the partial sums to any target.\n",
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