Difference between revisions of "Riemann zeta"
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(Created page with "Consider the function $\zeta$ defined by the series $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$ <div class="toccolours mw-collapsible mw-collapsed" style=...") |
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$$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$ | $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$ | ||
| + | ==Convergence== | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Proposition:</strong> If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges. | <strong>Proposition:</strong> If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges. | ||
Revision as of 00:24, 15 July 2014
Consider the function $\zeta$ defined by the series $$\zeta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{n^z}.$$
Convergence
Proposition: If $\mathrm{Re} \hspace{2pt} z > 1$, then the series defining $\zeta(z)$ converges.
Proof: █
