Fungrim entry: 22b67a

\sum_{n=1}^{\infty} \frac{1}{F_{n}^{2}} = \frac{5}{24} \left(\theta_{2}^{4}\!\left(0, \tau\right) - \theta_{4}^{4}\!\left(0, \tau\right) + 1\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=1}^{\infty} \frac{1}{F_{n}^{2}} = \frac{5}{24} \left(\theta_{2}^{4}\!\left(0, \tau\right) - \theta_{4}^{4}\!\left(0, \tau\right) + 1\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
`Pow`	${a}^{b}$	Power
`Fibonacci`	$F_{n}$	Fibonacci number
`Infinity`	$\infty$	Positive infinity
`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
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("22b67a"),
    Formula(Equal(Sum(Div(1, Pow(Fibonacci(n), 2)), For(n, 1, Infinity)), Where(Mul(Div(5, 24), Add(Sub(Pow(JacobiTheta(2, 0, tau), 4), Pow(JacobiTheta(4, 0, tau), 4)), 1)), 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