query: How to show that for any real numbers $\{a_n\}$ if $a_n\to 0$, then $\frac{a_1+\dots+a_n}{n}\to 0$.
positive_passages.docid: 193
positive_passages.text: On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $
negative_passages.docid: 49851
negative_passages.text: when the sum of some fractions be $1$
negative_passages.docid: 50087
negative_passages.text: Convergence of $a_{n+1}=\frac{1}{1+a_n}$
negative_passages.docid: 97656
negative_passages.text: Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
negative_passages.docid: 107824
negative_passages.text: Proof by induction that $a_0 = 1, a_1 = 1, a_n=2a_{n-1} + 3a_{n-2}$ satisfies $a_n = \frac12 (3^n) + \frac12 (-1)^n$
negative_passages.docid: 80960
negative_passages.text: If $a_n>0$ converges to $a>0$, then $(a_0 a_1\cdots a_n)^{\frac{1}{n}}$ converges to $a.$
negative_passages.docid: 37871
negative_passages.text: Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
negative_passages.docid: 43885
negative_passages.text: Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
negative_passages.docid: 3111
negative_passages.text: How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$
negative_passages.docid: 76925
negative_passages.text: Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $
negative_passages.docid: 63523
negative_passages.text: $|a_{n+1}-a_n| \leq \frac{1}{2} |a_{n}-a_{n-1}|$. Prove $(a_n)$ is Cauchy
negative_passages.docid: 93813
negative_passages.text: How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$
negative_passages.docid: 1275
negative_passages.text: Prove that $a_1+\cdots+a_n \geq n$ if $a_1$, $a_2$, ... $a_n$ are positive real numbers and their product is $1$
negative_passages.docid: 9662
negative_passages.text: For what real values of $a$ does $\sum_{n=1}^{\infty}{(\frac{an}{n +1})^n}$ converge?
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 63553
negative_passages.text: Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
negative_passages.docid: 33495
negative_passages.text: A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$