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Find $\displaystyle \lim_{n\rightarrow \infty }x_n$ : $$\left\{\begin{matrix}x_1=a>0\\ \\ x_{n+1}=\frac{2x_n\cdot \cos\left(\frac{\pi}{2^n+1}\right)}{x_n+1}\end{matrix}\right.$$

I have tried that : Let $a_n=\dfrac{1}{x_n}$ . So : $$a_{n+1}=\frac{1}{2\cos\left(\frac{\pi}{2^{n+1}}\right)}.a_n+\frac{1}{2\cos\left(\frac{\pi}{2^{n+1}}\right)}$$ So I tried to have geometric series: By let : $$a_{n+1}+f(n+1)=\frac{1}{2\cos\left(\frac{\pi}{2^{n+1}}\right)}(a_n+f(n))$$ So we must find one $f(n)$: $$\frac{f(n)}{2\cos\left(\frac{\pi}{2^{n+1}}\right)}-f(n+1)=\frac{1}{2\cos\left(\frac{\pi}{2^{n+1}}\right)}$$ As $$f(n)-f(n+1)\cdot 2\cos\frac{\pi}{2^{n+1}}=1$$

Can you give me the way to find one $f(n)$ sastisfied that ; or anyone has nice way to solve this problem

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up vote 1 down vote accepted


To show this, note that: first, $x_n\gt0$ for every $n\geqslant1$; second, $x_n\leqslant x^0_n$ for every $n\geqslant1$, where $x^0_1=a$ and $$ x^0_{n+1}=\frac{2x^0_n}{x^0_n+1}, $$ for every $n\geqslant1$; third, $x^0_n\to1$ when $n\to\infty$. Thus, $\limsup\limits_{n\to\infty}x_n\leqslant1$.

On the other hand, $\cos\left(\frac\pi{2^n+1}\right)\to1$ when $n\to\infty$. For every positive $b\lt1$, choose some positive integer $n(b)$ such that $\cos\left(\frac\pi{2^n+1}\right)\geqslant b$ for every $n\geqslant n(b)$ and define $(x^b_n)_{n\geqslant n(b)}$ by $x^b_{n(b)}=x_{n(b)}$ and $$ x^b_{n+1}=\frac{2bx^b_n}{x^b_n+1}, $$ for every $n\geqslant n(b)$. Then, $x_n\geqslant x^b_n$ for every $n\geqslant n(b)$ and, provided $b\gt\frac12$, $x^b_n\to x^b=2b-1$ when $n\to\infty$. Hence $\liminf\limits_{n\to\infty}x_n\geqslant x^b$. Since $x^b\to1$ when $b\to1$, this proves the result.

The same method shows that every sequence $(x_n)_{n\geqslant1}$ defined by $x_1=a\gt0$ and $$ x_{n+1}=\frac{c_nx_n}{x_n+1}, $$ for every $n\geqslant1$, with $c_n\gt0$ for every $n\geqslant1$ and $\lim\limits_{n\to\infty}c_n=c$ with $c\geqslant1$, converges to $c-1$. If $0\leqslant c\leqslant1$, $\lim\limits_{n\to\infty}x_n=0$.

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An incomplete answer that assumes the existence of the limit:

Since $\cos\frac{\pi}{2^n+1}$ goes to one for $n\to\infty$, the limit $x_\infty=\lim_{n\to\infty}x_n$ will, if it exists, satisfy the fixed-point equation $x_\infty=2x_\infty/(x_\infty+1)$. The only solutions of this equation are $0$ and $1$.

The derivative of $x\mapsto 2x\cos\frac{\pi}{2^n+1}/(x+1)$ at $x=0$ equals $2\cos\frac{\pi}{2^n+1}$ which is greater than one for $n>1$. Fixed-point theory then tells us that $x=0$ can not be the limit of your recursion. The derivative at $x=1$, in contrast, equals $1/2\cos\frac{\pi}{2^n+1}$ which is always positive and less than unity.

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