Take a look at this symbol:
$$ \pi=3 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \frac{(2k-1)^2} 6 $$
Does it look familiar to you? If so please help me!
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$\begingroup$ May I ask where you saw this? $\endgroup$– grayQuantMar 11, 2017 at 5:34
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2 Answers
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It is the notation for a continued fraction. In general: $$b_0 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \left(\frac{a_k}{b_k}\right)=b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}$$ Therefore, the continued fraction representation you have written above for $\pi$ is: $$\pi=3 + \underset{k=1}{\overset{\infty}{\large{\mathrm K}}} \frac{(2k-1)^2}{6}=3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \ddots\,}}}}$$ A proof of this result can be found on pages 399-401 of this document by Paul Loya.
answered Mar 9, 2017 at 22:29
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13$\begingroup$ An interesting aspect of this notation is that the apparent sequence of fractions $ \frac {a_k_} {b_k_} $ is not actually used. Given that, I find this notation rather unfortunate. $\endgroup$– PJTraillMar 10, 2017 at 9:05
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The given symbol is sometimes used for an infinite continued fraction, it seems to have been designed by Carl Friedrich Gauss.
answered Mar 9, 2017 at 22:14
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17$\begingroup$ A large Sigma for Summe (sum) by Euler, a large Pi for Product (product) by Gauß, a large Kappa for Kettenbruch (continued fraction) by Gauß as well. Neat! Obvious abbreviations are obvious. $\endgroup$– RolandMar 9, 2017 at 22:23
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