1. ## Proof of Convergence

I have a sequence of real numbers for which I wish to prove convergence.
$\displaystyle s_{n+1} = \sqrt[3]{s_{n}^{2} + s_{n-1} + 2}$ and $\displaystyle 0 < s_{0} < s_{1} < 2$

So, if the sequence $\displaystyle \{s_{n}\}$ converges to a limit $\displaystyle s$ then
$\displaystyle s^3 - s^2 - s - 2 = 0 \; \Rightarrow \; \left( s - 2 \right)\left( s^2 + s + 1 \right) = 0 \; \Rightarrow \; s = 2$
Also, we have that $\displaystyle t^3 - t^2 - t - 2 < 0 \; \Rightarrow \; t < s \forall t > 0$

But I have yet to prove convergence.
\displaystyle \begin{align*}s_{n+1}^3 - s^3 &= -s^3 + s_{n}^{2} + s_{n-1} + 2 \\ &= -\left( s^3 - s^2 - s - 2 \right) + \left( s_{n}^{2} - s^2 \right) + \left( s_{n-1} - s \right) \\ &= \left( s_{n}^{2} - s^2 \right) + \left( s_{n-1} - s \right) \\ \end{align*}
So if $\displaystyle s_{n-1}$ and $\displaystyle s_{n}$ are both less than $\displaystyle s$, then $\displaystyle s_{n+1} < s$ and our initial conditions give us $\displaystyle s_{n} < s \; \forall n \in \mathbb{N}$.
And thus the sequence $\displaystyle \{s_{n}\}$ is bounded above.

So now I need to prove that sequence $\displaystyle \{s_{n}\}$ is increasing.
\displaystyle \begin{align*} s_{n+1}^3 - s_{n}^3 &= -s_{n}^3 + s_{n}^{2} + s_{n-1} + 2 & \text{but} \\ s_{n} &< s \; \Rightarrow \; s_{n}^3 - s_{n}^{2} - s_{n} + 2 < 0 \\ \Rightarrow \; s_{n}^3 - s_{n}^{2} + 2 &< s_{n} & \text{so} \\ s_{n+1}^3 - s_{n}^3 &> s_{n-1} - s_{n} \\ \end{align*}
But that doesn't show that $\displaystyle s_{n+1}^3 - s_{n}^3 > 0$ as I require.

Can you help?

2. ## Re: Proof of Convergence

Have you learned that $\displaystyle f:\mathbb{R} \to \mathbb{R}$ is continuous if and only if $\displaystyle \{f(a_n)\}$ converges for all sequences $\displaystyle \{a_n\}$ which converge in $\displaystyle \mathbb{R}$? Well, you know that $\displaystyle f(x) = \sqrt[3]{x}$ is continuous. So, let $\displaystyle r_n = s_n^3$. If you can show that $\displaystyle r_n$ converges, then since $\displaystyle f$ is continuous, $\displaystyle f(r_n) = s_n$ converges, as well.

Anyway, to show convergence, you will want to end with something like:

Given $\displaystyle \varepsilon>0$, let $\displaystyle N$ be a positive integer such that $\displaystyle N > \text{expression}$ (where the expression is some function of $\displaystyle \varepsilon$).

Then show that for all $\displaystyle n > N$, $\displaystyle |s_n - 2| < \varepsilon$.

3. ## Re: Proof of Convergence

The text I'm looking at ("A First Course in Mathematical Analysis", Burkill) hasn't introduced the concept of a function at this point. So although your idea would work, I'm looking for something using much more basic ideas.

I don't intend to go into $\displaystyle \epsilon$ notation for this, since I have already shown that if the sequence converges, it must converge to 2. Given that, all I need is to show that it does converge. And the standard approach in the text is to show that the sequence is bounded (which I've done) and that it is monotonic. The combination of these results being sufficient for convergence. Another approach would be to show that $\displaystyle \left| s_n - 2 \right|$ converges to zero, but that doesn't look very practical here, the analysis of $\displaystyle s_{n+1}^3 - s^3$ proving only that the sequence is bounded.

4. ## Re: Proof of Convergence

Suppose $\displaystyle s_0 = 0.1, s_1 = 1.9$. Then $\displaystyle s_2 = \sqrt[3]{1.9^2 + 0.1 + 2} \approx 1.78736 < s_1$, so the sequence is not necessarily monotonically increasing. You might need to prove that $\displaystyle |s_n - 2|$ converges to zero.

5. ## Re: Proof of Convergence

You can, however, use the squeeze theorem. Let $\displaystyle r_{n+1} = \sqrt[3]{r_n^2 + r_{n-1} + 2}, r_0 = r_1 = 0$. Let $\displaystyle t_{n+1} = \sqrt[3]{t_n^2 + t_{n-1} + 2}, t_0 = t_1 = 2$. Obviously, $\displaystyle r_n < s_n < t_n$ for all $\displaystyle n$. It is pretty easy to show that $\displaystyle t_n = 2$ for all $\displaystyle n$. So, if you can show that $\displaystyle r_n$ converges to $\displaystyle 2$, then by the squeeze theorem, $\displaystyle s_n$ also converges to $\displaystyle 2$. Now, you can show that $\displaystyle r_n$ is bounded and monotonically increasing.

6. ## Re: Proof of Convergence

To show that $\displaystyle r_n$ converges, you know that $\displaystyle r_1 = 0, r_2 = \sqrt[3]{2} \approx 1.26, r_3 = \sqrt[3]{\sqrt[3]{4}+2} \approx 1.53$, so $\displaystyle r_1 < r_2 < r_3$. Assume that for all positive integers greater than or equal to 3 and up to $\displaystyle n$, $\displaystyle 2 > r_n > r_{n-1} > 1$ and $\displaystyle r_{n-1}>r_{n-2} > 0$ (which is true for $\displaystyle r_1,r_2,r_3$). Then $\displaystyle r_{n+1}^3 - r_n^3 = (r_n^2 + r_n + 2) - (r_{n-1}^2 + r_{n-2} + 2) = (r_n^2 - r_{n-1}^2) + (r_{n-1} - r_{n-2}) > 0$, so $\displaystyle r_{n+1} > r_n$. Hence, by mathematical induction, $\displaystyle r_n$ is monotone increasing. Then, since it is bounded, it converges. Finally, as you showed above, if it converges, it converges to 2. So, by the Squeeze Theorem, so too does $\displaystyle s_n$.

7. ## Re: Proof of Convergence

Haha, I hadn't realised that the initial values were sensitive like that. The original question has something like $\displaystyle s_0 = 1, s_1 = 1.3$

One thing I have started to think about is that
\displaystyle \begin{align*} s_{n+1}^3 - 2 &= s_{n}^2 + s_{n-1} + 2 - 2 \\ &= s_{n}^2 + s_{n-1} \\ \end{align*}
So $\displaystyle s_{n} > 0, s_{n-1} > 0$ (which is easily seen from the definition of $\displaystyle s_{n+1}$) gives $\displaystyle s_{n} \ge \sqrt[3]{2} \forall n > 0$

But then we also have
\displaystyle \begin{align*} s_{n+1}^3 - 4 &= s_{n}^2 + s_{n-1} + 2 - 4 \\ &= s_{n}^2 + s_{n-1} - 2 \\ \end{align*}
So $\displaystyle s_{n} > \sqrt[3]{2}, s_{n-1} > \sqrt[3]{2}$ gives $\displaystyle s_{n} \ge 4 \forall n > 2$.

And this is obviously something that can be repeated ad nauseam. I have yet to decide how useful it is though (if at all).

8. ## Re: Proof of Convergence

$r_{n+1}^3 - r_n^3 = (r_n^2 + r_n + 2) - (r_{n-1}^2 + r_{n-2} + 2) = (r_n^2 - r_{n-1}^2) + (r_{n-1} - r_{n-2}) > 0$
Ah! This is the trick I missed. Thanks a lot for your efforts.