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Let $a_1=-\frac32$, and $3a_{n+1}=2+a_n^3$. I need to show that $\displaystyle \lim_{n\to \infty} a_n = 1$.

I can show that the sequence is monotonically increasing and bounded as follows:

By the critical value method, $\frac13\left(2+x^3\right)>x$ for $x\in (-2,1)$. So, $a_n<a_{n+1}$ for all $n$. Also, if $f(x)=\frac13\left(2+x^3\right)$, then $f$ is increasing and $f(1)=1$. So, $a_n<1$ implies $a_{n+1}<1$. Hence the sequence is convergent, but I can't seem to find a good way to show that the $a_n$ are getting arbitrarily close to $1$.

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You've done most of the work already. Since $a_n$ is bounded and monotone increasing, then a limit $L$ exists. Further, since $a_{n+1}$ is a subsequence of $a_n$, then they have the same limit. That is, $\lim(a_n) = \lim(a_{n+1}) = L$. Taking a limit of both sides of your recursion, we get:

$$\lim(3a_{n+1}) = \lim(2+a_n^3)$$

And applying limit arithmetic to this, we arrive at:

$$3L = 2 + L^3$$

Now it's just a matter of solving for $L$.

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  • $\begingroup$ Nice answer. It's remarkable that the right idea is very often extremely simple. $\endgroup$ – Tim Raczkowski Dec 22 '14 at 3:44
  • $\begingroup$ Indeed! Usually all it takes is a little shift in perspective, which reminds me of this every time it happens to me: spikedmath.com/comics/557-Proofs-Part-1.png $\endgroup$ – Kaj Hansen Dec 22 '14 at 3:46
  • $\begingroup$ Yes. I know that feeling. :) $\endgroup$ – Tim Raczkowski Dec 22 '14 at 3:48
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Now that you know that your iteration is convergent, you have that any limit must be a fixed point of the iteration function. That is, if $L$ is the limit, then

$$3L = 2+L^3$$

or

$$L^3-3L+2=0.$$

This comes from taking limits on both sides. With some guess and check, you can factor this as

$$(L-1)^2(L+2)=0.$$

So the limit is either $1$ or $-2$. But the sequence is increasing from $-3/2$, so the limit is $1$.

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$\text{ Since}$ $a_n > a_1 = -\dfrac{3}{2}, \forall n \geq 1 \Rightarrow \displaystyle L=\lim_{n\to \infty} a_n \geq a_1 = -\dfrac{3}{2}$. But $L^3 - 3L + 2 = 0 \to (L-1)^2(L+2) = 0 \to L = 1 \text{ or } L = -2$. $\text{ Since}$ $L > -2$, $L = 1$.

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