How to show that for sequence $x_1=2$, and $x_{n+1}=\frac12\big(x_n+\frac2{x_n}\big)$; for all $n$, $(x_n)^2>2$?
142
Showing the sequence converges to the square root
34025
Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
132947
How to calculate the limit of $\frac{a_n}{n^2}$ for the sequence $a_{n+1}=a_n+\frac{2 a_{n-1}}{n+1}$?
95095
$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.
123523
Showing $\left(1+\frac{1}{n}\right)^n$ is an increasing sequence by comparing the binomial expansions term by term
156367
How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$
143236
$ S_{n}=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+...........+\frac{x^{2^{n}}}{(x+1)(x^2+1)...(x^{2^{n}}+1)}$
94409
showing that the sequence $a_n=1+\frac{1}{2}+...+\frac{1}{n} - \log(n)$ converges
63553
Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
145863
sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
135018
Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
78472
Sequence is periodic $x_{n+2}=|x_{n+1}-x_{n-1}|$
22077
Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$
121606
How do I prove that the limit of the sequence exists: $a_n = \frac1n +\frac 1{n+1} + \frac1{n+2} + \dots + \frac1{2n}$?
44622
If $F_n$ is the Fibonacci sequence, show that $F_n < \left(\frac 74\right)^n$ for $n\geq 1.$
111353
Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
122820
Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$