Sum to Infinity of Reciprocal of n by 2n Choose n

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Theorem

\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n \dbinom {2 n} n}\) \(=\) \(\ds \frac {\pi \sqrt 3} 9\)
\(\ds \) \(\approx\) \(\ds 0 \cdotp 60459 \, 97880 \, 7807 \ldots\)

This sequence is A073010 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

By Sum to Infinity of 2x^2n over n by 2n Choose n:

\(\ds \frac {2 x \arcsin x} {\sqrt {1 - x^2} }\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}\)
\(\ds \frac {2 \paren {\frac 1 2} \arcsin \frac 1 2} {\sqrt {1 - \paren {\frac 1 2}^2} }\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {2 \paren {\frac 1 2} }^{2 n} } {n \dbinom {2 n} n}\) substituting $x = \dfrac 1 2$
\(\ds \frac {\frac \pi 6} {\sqrt {\frac 3 4} }\) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n \dbinom {2 n} n}\) Sine of $30^\circ$
\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n \dbinom {2 n} n}\) \(=\) \(\ds \frac {\pi \sqrt 3} 9\)

$\blacksquare$


Also see


Sources

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