# value of $\lim\limits_{n\rightarrow \infty}\left(a_{n+2}-a_{n}\right)$ is

Let $$a_{1}=a$$ and $$a_{n+1}=\cos(a_{n})\;\forall \;n\;\in \mathbb{N}.$$

Then $$\lim\limits_{n\rightarrow \infty}(a_{n+2}-a_{n})$$ is

Try: $$a_{n+2}=\cos(a_{n+1})=\cos(\cos(a_{n}))=\cos(\cos(\cos (a_{n-1})))=\cdots \cdots \cos(\cos\cos\cos(\cdots \cdots \cos(a)))))))$$

Did not know how can i solve it, could some help me , Thanks

• Try to show that $\lim\limits_{n\rightarrow\infty}a_n$ exists, first (e.g. here). Then the limit of the difference is the difference of the limits. Or the sequence is Cauchy. – rtybase Feb 26 at 13:45

Hint: Apply Banach's fixed point theorem to $$\varphi(x) = \cos(\cos(x))$$.

• Did not understand , Can you explain me. – DXT Feb 26 at 13:52
• @DXT What exactly did you not understand? – Klaus Feb 26 at 13:53
• I mean Branch Fixed point Theorem – DXT Feb 26 at 13:57
• @DXT If you were given this exercise, I am sure you had Banach's fixed point theorem in class. You can read it up here for example: en.wikipedia.org/wiki/Banach_fixed-point_theorem ($T$ is $\varphi$ and $X$ is any sufficiently large interval here) – Klaus Feb 26 at 14:00

Hints:

• $$\cos: [0,1] \rightarrow [0,1]$$
• $$|\cos'(x)| = |\sin x| \leq \sin 1 < \frac{9}{10}\Rightarrow \cos$$ is contractive on $$[0,1]$$
• $$\Rightarrow$$ $$a_{n+1} = \cos a_n$$ converges to the only fixpoint in $$[0,1]$$.

Now, reason why irrespective of the starting value $$a$$ the sequence $$a_n$$ will have to fall into $$[0,1]$$ "earlier or later".

• $\cos{x}$ is $[-1,1]\rightarrow[-1,1]$. – rtybase Feb 26 at 13:55
• @rtybase : $\cos (-1) = \cos 1 \in [0,1]$. So, my one is a bit "narrower". – trancelocation Feb 26 at 13:59
• Say initial condition is $a_0=3$, then $\cos(a_0)=-0.9899924...$. It's still fine to consider $\cos$ as a contraction mapping on $[-1,1]$, it has one fixed point anyway. – rtybase Feb 26 at 14:03
• @rtybase In any case, for a sufficiently large $n$ (actually quite small), you have that $a_n\in[0,1]$. – egreg Feb 26 at 14:05
• @egreg I know. It's not me who should provide these details. And it's really easy to update the answer and close this little gap. – rtybase Feb 26 at 14:06