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  "body": "\n## Example\n\nThe harmonic series is the series\n$$\\sum_{n=1}^{\\infty} \\frac{1}{n} = 1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots.$$\nIts terms tend to $0$, but the series is a [[divergent series|0NBRP6093V]].\nIts partial sums are the [[harmonic numbers|0D2Y6G4C4S]] $H_N = \\sum_{n=1}^{N} 1/n$, and they tend to $+\\infty$.\n\n## Remarks\n\nThe node [[harmonic series diverges|0E9M24KS9J]] gives a proof.\nIt shows that the converse of the [[term test|0VVY4H92G2]] is false.\nThe partial sums grow very slowly, like $\\ln N$. The node [[harmonic numbers grow like log n|05S2GSPMGA]] makes this precise.\nThe name comes from music. The wavelengths of the overtones of a string are $1/2, 1/3, 1/4, \\dots$ of the wavelength of the string.\n\n## Explanations\n\n### L1 human\n\nThe terms $1, 1/2, 1/3, \\dots$ become small, but not fast enough.\nYou can group them into blocks that each add up to at least $1/2$, so the total grows without bound.\n",
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