Show that the sequence $a_n=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ is rising and is unbounded.
14031
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
25921
Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
155695
The convergence of a sequence
107399
A sequence and convergence
5797
Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$
27410
If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
14489
Showing that a sequence is monotone and thus convergent
152129
$a_1=3$ and $a_{n+1}=\frac{a_n}{2} + \frac{1}{a_n}$. Show that it monotonically decreases and find the limit.
31930
Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.
112663
Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity
147661
Show that {$x_n$} is a convergent sequence if $d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n)$
19657
Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
38002
how to prove that this positive sequence converges to 1
110680
Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$
150000
Convergence of the sequence $\frac{1}{n\sin(n)}$
143227
Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
30471
Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.