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  "body": "\n## Statement\n\nThe partial sums $H_N = \\sum_{n=1}^{N} \\frac{1}{n}$ of the [[harmonic series|0CXTMAY5PX]] tend to $+\\infty$ as $N \\to \\infty$.\nIn particular, the harmonic series is a divergent series.\n\n## Remarks\n\nThe classical proof groups the terms in blocks from $2^{k-1} + 1$ to $2^k$. Each block has $2^{k-1}$ terms, each at least $1/2^k$, so each block sums to at least $1/2$. Thus $H_{2^k} \\ge 1 + k/2$.\nA second proof uses the [[integral test|019QZ6RRVH]]: $H_N \\ge \\int_1^{N+1} dx/x = \\ln(N + 1)$.\nMathlib states it as the limit of $\\sum_{n \u003c N} 1/(n+1)$, in `Real.tendsto_sum_range_one_div_nat_succ_atTop`.\n\n## History\n\nNicole Oresme proved the divergence in the 14th century, with the grouping argument. Pietro Mengoli and Jacob Bernoulli gave other proofs in the 17th century.\n\n## Explanations\n\n### L1 human\n\nGroup the terms: $1/3 + 1/4 \\ge 1/2$, then $1/5 + \\cdots + 1/8 \\ge 1/2$, then $1/9 + \\cdots + 1/16 \\ge 1/2$, and so on.\nThere are infinitely many such groups, so the sum exceeds each bound.\n",
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