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  "body": "\n## Statement\n\nFor every real $x$ with $-1 \u003c x \\le 1$,\n$$\\ln(1 + x) = \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{n} x^n = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\cdots.$$\n\n## Remarks\n\nFor $|x| \u003c 1$, integrate the [[geometric series|01X5CVV1Q5]] $\\frac{1}{1 + t} = \\sum (-t)^n$ term by term from $0$ to $x$.\nAt $x = 1$ the series converges by the alternating series test. [[Abel's theorem|0S4EMDQS2F]] gives $\\ln 2 = 1 - \\frac{1}{2} + \\frac{1}{3} - \\cdots$.\nMathlib states the form $\\sum_{n \\ge 0} \\frac{x^{n+1}}{n+1} = -\\ln(1 - x)$ as `Real.hasSum_pow_div_log_of_abs_lt_one`.\n\n## History\n\nJohannes Hudde found the series in 1656 and Isaac Newton in 1665, but neither of them published it.\nNicholas Mercator found it independently and used it in his book Logarithmotechnia of 1668.\n\n## Explanations\n\n### L1 human\n\nThe logarithm of $1 + x$ is a power series in $x$, for $x$ in $(-1, 1]$.\nAt $x = 1$ it gives the alternating harmonic series, with sum $\\ln 2$.\n",
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