Fungrim entry: 344963

\sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n + 1}}{F_{n} F_{n + 1}} = \frac{\sqrt{5} - 1}{2}

References:

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

TeX:

\sum_{n=1}^{\infty} \frac{{\left(-1\right)}^{n + 1}}{F_{n} F_{n + 1}} = \frac{\sqrt{5} - 1}{2}

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
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("344963"),
    Formula(Equal(Sum(Div(Pow(-1, Add(n, 1)), Mul(Fibonacci(n), Fibonacci(Add(n, 1)))), For(n, 1, Infinity)), Div(Sub(Sqrt(5), 1), 2))),
    References("J. M. Borwein and P. B. Borwein. Pi and the AGM. Wiley, New York, 1987."))

Topics using this entry

Fibonacci numbers

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