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Why do the sum of such series tend to involve $π$,which belongs to a circle? How does a series as below have anything to do with a circle?

$$ \begin{array}{l} \frac{1}{1^6}+\frac{1}{3^{6}}+\frac{1}{5^{6}}+\cdots=\frac{\pi^{6}}{960} \\ \frac{1}{1^{4}}+\frac{1}{3^{4}}+\frac{1}{5^{4}}+\cdots=\frac{\pi^{4}}{96} \end{array} $$

RobPratt
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Kashmiri
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  • https://math.stackexchange.com/questions/689315/interesting-and-unexpected-applications-of-pi – Anton Vrdoljak Aug 22 '20 at 11:01
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    Notice $\sum_{n \geq 1} \frac{1}{(2n-1)^k}=(1-\frac{1}{2^k})\sum_{n \geq 1} \frac{1}{n^k}$ and presence of $\pi$ in $\sum_{n \geq 1} \frac{1}{n^k}$ for even $k$ is discussed in question referenced above (see also https://en.wikipedia.org/wiki/Basel_problem). – Sil Aug 22 '20 at 11:09
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    There's more to $\pi$ than circles! – Angina Seng Aug 22 '20 at 11:34
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    @Angina, pi is imo basically related to a circle. That's it's first definition isn't it? – Kashmiri Aug 22 '20 at 11:39
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    This isn't completely related to the question, but you would enjoy this playlist: https://www.youtube.com/watch?v=d-o3eB9sfls&list=PLZHQObOWTQDMVQcT3414TcPMeEYf_VtPM – Sage Stark Aug 22 '20 at 11:43
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    That's a little like saying, why is carbon involved in all these chemicals that have nothing to do with coal, Yasir. It may be your opinion that carbon is basically related to coal, but neither carbon nor $\pi$ cares what your opinions are. – Gerry Myerson Aug 22 '20 at 12:34
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    @YasirSadiq Historically Yes. In modern mathematics you define $\pi$ as a certain zero of the cosine, and you define the cosine as a certain solution of a certain differential equation. No more circles involved at this stage. – M. Winter Aug 22 '20 at 14:15
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    To emphasize, some of the videos in the 3Blue1Brown playlist that @Pendronator linked directly address ways to connect a circle to the "Basel problem" that Sil mentioned, which is very closely related to the series presented in the question. So I feel like those videos are a good answer (but not perfect in that they don't focus much on the higher powers in the question). – Mark S. Aug 22 '20 at 16:07
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    On the plus side don't you find it interesting enough that $\pi$ has a humble beginning with circles and presents itself in many diverse areas (including your infinite series)? – Paramanand Singh Aug 23 '20 at 02:59

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