# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC