Haven't learned calculus yet but I need this for a proof for Fibonacci numbers and its limit In an infinite series of Fibonacci numbers, is this always true $$\frac{F_{n}}{F_{n+1}}>\frac{F_{n-1}}{F_n}$$? Can you make an argument that in an infinite convergent series, eventually that will be false?

hint :We know **$\color{red} {F_{n+1}=F_n+F_{n-1}}\\ \to \color{red} {F_{n}=F_{n+1}-F_{n-1}}$
$$\frac{F_{n+1}-F_{n-1}}{F_{n+1}}>\frac{F_{n-1}}{F_{n+1}-F_{n-1}} $$
or see this $$\begin{bmatrix}f_{n+1} & f_n \\f_n & f_{n-1} \end{bmatrix}=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}^n \to \det(\begin{bmatrix}f_{n+1} & f_n \\f_n & f_{n-1} \end{bmatrix})=\det(\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}^n)\\f_{n+1}f_{n-1}-f_n.f_n=(-1)^n\\n=2k \to f_{n+1}f_{n-1}-f_n.f_n=+1 \\f_{n+1}f_{n-1}>f_n.f_n\\ \frac{f_{n+1}}{f_{n}}>\frac{f_{n}}{f_{n-1}}\\
n=2k+1 
\to f_{n+1}f_{n-1}-f_n.f_n=-1 \to f_{n+1}f_{n-1}<f_n.f_n \\ \to  \frac{f_{n+1}}{f_{n}}<\frac{f_{n}}{f_{n-1}} \to \\
\color{red} {\frac{f_{n}}{f_{n+1}}>\frac{f_{n-1}}{f_{n}}}$$ this is your case  when $n$ is odd
