query: Provided that $\lim \limits_{n\to \infty} u_n=l$, prove that $\lim \limits_{n\to \infty} \frac{\sum_{k=1}^nu_k}{n}=l$
positive_passages.docid: 154553
positive_passages.text: If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$?
negative_passages.docid: 17157
negative_passages.text: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
negative_passages.docid: 44288
negative_passages.text: Is my proof of convergence for $\sum\limits^{\infty}_{n=1} \frac {2^n} {n^n}$ true?
negative_passages.docid: 10211
negative_passages.text: Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$
negative_passages.docid: 70491
negative_passages.text: If $\sum_{n=1}^\infty a_n$ converges, prove that $\lim_{n\to \infty} (1/n) \sum_{k=1}^n ka_k = 0$.
negative_passages.docid: 149586
negative_passages.text: If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
negative_passages.docid: 11879
negative_passages.text: If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
negative_passages.docid: 128882
negative_passages.text: Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$
negative_passages.docid: 11435
negative_passages.text: Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges
negative_passages.docid: 73812
negative_passages.text: Does $\lim \limits_ {n\to \infty}\ \frac{n}{n+1}$ equal $1$?
negative_passages.docid: 11539
negative_passages.text: Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$
negative_passages.docid: 15292
negative_passages.text: Elementary proof for $\lim\limits_{n \to\infty}\frac{n!e^n}{n^n} = +\infty$
negative_passages.docid: 35325
negative_passages.text: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
negative_passages.docid: 143227
negative_passages.text: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
negative_passages.docid: 59162
negative_passages.text: Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
negative_passages.docid: 155333
negative_passages.text: Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
negative_passages.docid: 90773
negative_passages.text: $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{k^{2}+1}$