Sum to Infinity of Reciprocals of Central Binomial Coefficients

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Theorem

\(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\dbinom {2 n} n}\) \(=\) \(\ds \frac {2 \pi \sqrt 3 + 9} {27}\)
\(\ds \) \(\approx\) \(\ds 0 \cdotp 73639 \, 98587 \ldots\)

This sequence is A073016 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:

\(\ds \sum_{n \mathop = 1}^\infty \frac {\paren {2 x}^{2 n} } {n \dbinom {2 n} n}\) \(=\) \(\ds \frac {2 x \arcsin x} {\sqrt {1 - x^2} }\)
\(\ds \leadsto \ \ \) \(\ds \sum_{n \mathop = 1}^\infty \frac {2 n \paren {2^{2 n} x^{2 n - 1} } } {n \dbinom {2 n} n}\) \(=\) \(\ds \frac {\paren {2 \arcsin x + \frac {2 x} {\sqrt {1 - x^2} } } \sqrt {1 - x^2} - 2 x \arcsin x \paren {\frac 1 2} \paren {-2 x} \paren {1 - x^2}^{-\frac 1 2} } {1 - x^2}\) differentiating both sides with respect to $x$
\(\ds \) \(=\) \(\ds \frac {2 x + 2 \arcsin x \sqrt {1 - x^2} + 2 x^2 \arcsin x \paren {1 - x^2}^{-\frac 1 2} } {1 - x^2}\)
\(\ds \) \(=\) \(\ds \frac {2 x} {1 - x^2} + \frac {2 \arcsin x \paren {\paren {1 - x^2} + x^2} } {\paren {1 - x^2}^{\frac 3 2} }\)
\(\ds \) \(=\) \(\ds \frac {2 x} {1 - x^2} + \frac {2 \arcsin x} {\paren {1 - x^2}^{\frac 3 2} }\)
\(\ds \leadsto \ \ \) \(\ds \sum_{n \mathop = 1}^\infty \frac 4 {\dbinom {2 n} n}\) \(=\) \(\ds \frac {2 \paren {\frac 1 2} } {1 - \paren {\frac 1 2}^2} + \frac {2 \arcsin \paren {\frac 1 2} } {\paren {1 - \paren {\frac 1 2}^2}^{\frac 3 2} }\) substituting $x = \dfrac 1 2$
\(\ds \) \(=\) \(\ds \frac 1 {\frac 3 4} + \frac {2 \paren {\frac \pi 6} } {\paren {\frac 3 4}^{\frac 3 2} }\) Sine of $30 \degrees$
\(\ds \) \(=\) \(\ds \frac 4 3 + \frac \pi 3 \times \frac 8 {3 \sqrt 3}\)
\(\ds \) \(=\) \(\ds \frac {36 + 8 \pi \sqrt 3} {27}\)
\(\ds \leadsto \ \ \) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {\dbinom {2 n} n}\) \(=\) \(\ds \frac {9 + 2 \pi \sqrt 3} {27}\)

$\blacksquare$


Also see


Sources

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