Pell's function

Let $\displaystyle \lim_{n\rightarrow \infty} \dfrac {P_n}{P_{n-1}} = \delta _s$

We have $P_n = 2P_{n-1} + P_{n-2}$

Dividing both sides by $P_{n-1},$ $\dfrac {P_n}{P_{n-1}} = 2 + \dfrac {P_{n-2}}{P_{n-1}}$

$\Rightarrow \delta _s = 2 + \dfrac {1}{\delta _s}$

Solving we get , $\displaystyle \lim_{n\rightarrow \infty}\dfrac {P_n}{P_{n-1}} = \delta _s = \sqrt 2 + 1 =\boxed {2.4142..}$

Infact you can generalize a sequence which can be defined as, $P_n = NP_{n-1} + P_{n-2}$

Doing the same as above you can get the ratio as, $(\delta _s)_N = \dfrac {N + \sqrt{N^2 + 4}}{2}$

Input $N=2$ gives you the Silver Ratio, $N=1$ gives you Golden Ratio . Actually there are number of such ratios grouply called as The Mettalic Ratios.

Relevant wiki: Linear Recurrence Relations - Calculating Initial Terms

The characteristic polynomial of the linear recurrence relation of Pell numbers is:

$\begin{aligned} r^2 & = 2r + 1 \\ r^2 - 2r - 1 & = 0 \\ \implies r & = 1 \pm \sqrt 2 \\ \implies P_n & = c_1 (1+\sqrt 2)^n + c_2(1-\sqrt 2)^n \\ P_0 & = c_1+c_2 = 0 & \small \color{#3D99F6} \implies c_2 = - c_1 \\ P_1 & = c_1(1+\sqrt 2) - c_1(1-\sqrt 2) = 1 & \small \color{#3D99F6} \implies c_1 = \frac 1{2\sqrt 2} \\ \implies P_n & = \frac {(1+\sqrt 2)^n - (1-\sqrt 2)^n}{2\sqrt 2} \end{aligned}$

Therefore,

$\begin{aligned} \lim_{n \to \infty} \frac {P_n}{P_{n-1}} & = \lim_{n \to \infty} \frac {(1+\sqrt 2)^n - (1-\sqrt 2)^n}{(1+\sqrt 2)^{n-1} - (1-\sqrt 2)^{n-1}} & \small \color{#3D99F6} \text{Divide up and down by }(1+\sqrt 2)^{n-1} \\ & = \lim_{n \to \infty} \frac {1+\sqrt 2-(1-\sqrt 2)\left(\frac {1-\sqrt 2}{1+\sqrt 2}\right)^{n-1}}{1-\left(\frac {1-\sqrt 2}{1+\sqrt 2}\right)^{n-1}} \\ & = 1 + \sqrt 2 \approx \boxed{2.4142} \end{aligned}$