Given the recursive sequence:$x_1=1$ and $x_{n+1}=\frac{1}{3}(x_n+\frac{3}{x_n})$, prove that it is a Cauchy sequence and find its limit.
142
Showing the sequence converges to the square root
113340
Show that the recursive sequence is bounded
20909
For each positive integer $n$, let $x_n=1/(n+1)+1/(n+2)+\cdots+1/(2n)$. Prove that the sequence $(x_n)$ converges.
153736
Question about convergence of a simple recursive sequence
29138
If $\{x_n\}$ satisfies that $x_{n+1} - x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?
70561
limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$
19657
Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
154694
Is being a Cauchy sequence equivalent to $ \lim_{n\to+\infty}d(x_{n+k},x_n)=0$ for every $k$?
130877
Boundedness and Cauchy Sequence: Is a bounded sequence such that $\lim(a_{n+1}-a_n)=0$ necessarily Cauchy?
99933
Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$
18675
$\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$
112595
Prove that the limit of this sequence is $\frac{1}{2}(1 + \sqrt{5})$
147661
Show that {$x_n$} is a convergent sequence if $d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n)$
3211
The sequence $x_{n+1}=\frac{x_n}{2}-\frac{2}{x_n}, x_0>0$ is bounded?
46830
Prove that the sequence $a_{n+1} =\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right)$ is convergent and find its limit
37906
Prove $s_n$ is a Cauchy sequence for $|s_{n+1}-s_n|\lt2^{-n}$
33495
A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$