Fungrim entry: da1873

\sum_{n=0}^{\infty} \frac{1}{F_{2 n + 1}} = \frac{\sqrt{5}}{4} \theta_{2}^{2}\!\left(0, \tau\right)\; \text{ where } \tau = \frac{1}{\pi i} \log\!\left(\frac{3 - \sqrt{5}}{2}\right)

References:

J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987.

TeX:

\sum_{n=0}^{\infty} \frac{1}{F_{2 n + 1}} = \frac{\sqrt{5}}{4} \theta_{2}^{2}\!\left(0, \tau\right)\; \text{ where } \tau = \frac{1}{\pi i} \log\!\left(\frac{3 - \sqrt{5}}{2}\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Fibonacci`	$F_{n}$	Fibonacci number
`Infinity`	$\infty$	Positive infinity
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`JacobiTheta`	$\theta_{j}\!\left(z , \tau\right)$	Jacobi theta function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Log`	$\log(z)$	Natural logarithm

Source code for this entry:

Entry(ID("da1873"),
    Formula(Equal(Sum(Div(1, Fibonacci(Add(Mul(2, n), 1))), For(n, 0, Infinity)), Where(Mul(Div(Sqrt(5), 4), Pow(JacobiTheta(2, 0, tau), 2)), Equal(tau, Mul(Div(1, Mul(Pi, ConstI)), Log(Div(Sub(3, Sqrt(5)), 2))))))),
    References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987."))

Topics using this entry

Fibonacci numbers