Difference between revisions of "Catalan's constant"
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$$G=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^2} = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 \ldots.$$ | $$G=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^2} = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 \ldots.$$ | ||
This means that Catalan's constant can be expressed as $\beta(2)$ where $\beta$ is the [[Dirichlet beta function]]. | This means that Catalan's constant can be expressed as $\beta(2)$ where $\beta$ is the [[Dirichlet beta function]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Proposition:</strong> The following formula holds: | ||
| + | $$K=\beta(2),$$ | ||
| + | where $K$ is [[Catalan's constant]] and $\beta$ denotes the [[Dirichlet beta]] function. | ||
| + | where | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 01:13, 21 March 2015
Catalan's constant is $$G=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^2} = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 \ldots.$$ This means that Catalan's constant can be expressed as $\beta(2)$ where $\beta$ is the Dirichlet beta function.
Properties
Proposition: The following formula holds: $$K=\beta(2),$$ where $K$ is Catalan's constant and $\beta$ denotes the Dirichlet beta function. where
Proof: proof goes here █
