Prove that $a_{n+1}=a_n^2+a_n^3$ $a_1=\frac{1}{2}$ converges (recursion sequence

real-analysis

Prove that $a_{n+1}=a_n^2+a_n^3$ and $a_1=\frac{1}{2}$ converges (I just need this part and then I will continue the rest but I will put the full question anyway even if it is not needed)

and prove $b_{n+1}=b_n^2+b_n^3$ and $b_1=-1$ converges as well

let $c_n$ be the sequence $
c_n=
\begin{cases}
a_n&\text{if}\, n _{even}\\
b_n&\text{if}\, n_{odd}\\
\end{cases}
$
does $c_n$ converge?

We know that $a_1=\frac{1}{2}$ and $a_2=\frac{3}{8}$ , $a_3=\frac{99}{512}$ so the impression is that it is a decreasing sequence.

I did not prove that it is well defined because I think that it is trivial and there is not much to prove in this

I tried getting an impression of what the limit will be and did solved the following , assume that $a_n$ converges then $L=L^2+L^3$ and after solving I got $L_1=0$ ,$L_2=\frac{1+ \sqrt5}{2}$ , $L_3=\frac{1-\sqrt5}{2}$
I am not sure if this is right but from this we can get $0 \leq a_n \leq \frac{1+ \sqrt5}{2}$

I believe that here I should prove that the sequence is bounded and monotone because we know that every sequence that is monotone and bounded converges but how can I prove that it is decreasing and bounded ?

I tried induction and I did not know how to continue

Induction basis for $n=1$ – $a_1=\frac{1}{2}$ and $a_2=\frac{3}{8}$ as mentioned above

Induction proposition – for $n=k \geq 2$ – to show that the sequence is decreasing $a_{k-1} \geq a_k$

Induction step prove for $n=k+1$ – $a_k \geq a_{k+1}$

so we need to prove $a_k \geq a_{k+1}$ and from what we assume in the induction $a_{k-1} \geq a_k$ so $a_{k-1} \geq a_k \geq a_{k+1}$ $\to$ $a_{k-1} \geq a_{k+1}=a_k^2 +a_k^3$ but how do I continue from here (if it is right)?

Thanks for any help and tips hopefully the translations are understandable

Edit – I now believe that what I did with the $"L"$ is not correct because if a sequence is decreasing then it is bounded from above by $a_1$ and not the $L$ that I found

Best Answer

If all you need to prove is convergence for $a_1=1/2$, show that the sequence is monotonic and bounded.

  • With $a_1$ positive, every term is forced positive, giving a lower bound of $0$.

  • Render $a_{n+1}/a_n=a_n(1+a_n)$, which is $<1$ for $0\le a_n \le 1/2$.

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