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?

$$\frac{F_{n}}{F_{n+1}}>\frac{F_{n-1}}{F_n}$$
is equivalent to
$$F_{n}^{2} - F_{n-1} F_{n+1} > 0.$$
Using
\begin{align}
5 \, F_{n}^{2} &= L_{2n} - 2 (-1)^{n} \\
5 \, F_{n-1}F_{n+1} &= L_{2n} + 3 (-1)^{n},
\end{align}
where $L_{n}$ are the Lucas numbers, then
$$-5 (-1)^n > 0$$
is the result. Only the odd values of $n$ make the given inequality true, ie
$$\frac{F_{2n+1}}{F_{2n+2}}>\frac{F_{2n}}{F_{2n+1}}$$
