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  "body": "\n## Example\n\nFor a complex number $s$ with real part $\\sigma = \\operatorname{Re} s$, consider the series\n$$\\zeta(s) = \\sum_{n=1}^{\\infty} \\frac{1}{n^s}, \\qquad n^{-s} = e^{-s \\log n}.$$\nIt is the [[Dirichlet series|HN0AXS6VSG]] of the constant arithmetic function $a(n) = 1$.\nIts terms satisfy $\\abs{n^{-s}} = n^{-\\sigma}$.\nBy the [[p-series test|0ZNSSK8ZK1]], $\\sum_n n^{-\\sigma}$ converges exactly when $\\sigma \u003e 1$.\nSo the series converges [[absolutely|0W3K3QPEWT]] for every $s$ with $\\operatorname{Re} s \u003e 1$.\n\nFix $\\sigma_0 \u003e 1$; on the half-plane $\\operatorname{Re} s \\ge \\sigma_0$ the bound $\\abs{n^{-s}} \\le n^{-\\sigma_0}$ holds.\nSo the [[Weierstrass M-test|0NX5NWAFC9]], in its form for complex-valued functions, shows that the series converges uniformly there.\nOn the whole half-plane $\\operatorname{Re} s \u003e 1$ the convergence is not uniform.\nIndeed, for real $s \\to 1^+$ the tail $\\sum_{n \u003e N} n^{-s}$ tends to $\\sum_{n \u003e N} n^{-1} = \\infty$.\nSo for every $N$ the tail is unbounded on the half-plane.\nAt $s = 1$ the series is the [[harmonic series|0CXTMAY5PX]], which diverges.\n\n## Remarks\n\nThe sum of this series on $\\operatorname{Re} s \u003e 1$ is the [[Riemann zeta function|1XWBY4N0VT]], which extends beyond the half-plane by analytic continuation.\nAt $s = 2$ the series gives $\\zeta(2) = \\pi^2/6$, the [[Basel problem|0DMY75D6FV]].\n\n## Explanations\n\n### L1 human\n\nAdd up $1$, then $1/2^s$, then $1/3^s$, and so on.\nFor $s = 2$ this is $1 + 1/4 + 1/9 + \\cdots$, which adds up to $\\pi^2/6$.\nFor $s = 1$ the terms shrink too slowly and the sum grows without bound.\nThe series works exactly when the real part of $s$ is larger than $1$, and it defines the zeta function there.\n",
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