# Fungrim entry: 05209f

$\sum_{n=0}^{\infty} F_{n} {z}^{n} = \frac{z}{1 - z - {z}^{2}}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < \varphi - 1$
TeX:
\sum_{n=0}^{\infty} F_{n} {z}^{n} = \frac{z}{1 - z - {z}^{2}}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|z\right| < \varphi - 1
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Fibonacci$F_{n}$ Fibonacci number
Pow${a}^{b}$ Power
Infinity$\infty$ Positive infinity
CC$\mathbb{C}$ Complex numbers
Abs$\left|z\right|$ Absolute value
GoldenRatio$\varphi$ The golden ratio (1.618...)
Source code for this entry:
Entry(ID("05209f"),
Formula(Equal(Sum(Mul(Fibonacci(n), Pow(z, n)), For(n, 0, Infinity)), Div(z, Sub(Sub(1, z), Pow(z, 2))))),
Variables(z),
Assumptions(And(Element(z, CC), Less(Abs(z), Sub(GoldenRatio, 1)))))

## 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