prove: $\lim\limits_{t \to 0} (\frac{1}{n}\sum_{k=1}^{n}x_{k}^{t})^{\frac{1}{t}} = (x_{1}\cdots x_{n})^{\frac{1}{n}} $
66089
Using AM-GM to prove $\lim_{p\to 0}$ of the $p$-mean is equal to the GM.
21504
Showing that $\lim\limits_{n\to\infty}\sum^n_{k=1}\frac{1}{k}-\ln(n)=0.5772\ldots$
17157
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
87116
Prove $ \lim_{n\to \infty}\frac{n}{n+1} = 1$ using Cauchy (epsilon) definition of a limit of a sequence
89263
Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$
72960
How to prove this limit composition theorem?
2449
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
106281
Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$
121606
How do I prove that the limit of the sequence exists: $a_n = \frac1n +\frac 1{n+1} + \frac1{n+2} + \dots + \frac1{2n}$?
143227
Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
80277
How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
42519
$\lim_{n \rightarrow \infty} \sum^n_{k=1} \frac{1}{k+n}$
5436
How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \ $?
108405
Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$
156367
How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$
56902
Evaluate $\lim_{n \to \infty} \frac{1}{n}[(n +1)(n+2) \cdots (n+n)]^{\frac{1}{n}}$
95095
$x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.