query: If $\lim_{n \to\infty}a_n=a$ then $\lim_{n\to \infty}\frac{a_1+a_2+...+a_n}{n}=a$
positive_passages.docid: 13701
positive_passages.text: Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means
negative_passages.docid: 111211
negative_passages.text: Does $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ imply that $a_n$ convergent?
negative_passages.docid: 145772
negative_passages.text: Proving $\liminf_{n\to\infty}(-a_n)=-\limsup_{n\to\infty}(a_n)$
negative_passages.docid: 3527
negative_passages.text: Finding $\lim_{n \to \infty} \prod_{r=1}^n (1+ \frac{1}{a_r})$
negative_passages.docid: 7815
negative_passages.text: Prove that $\lim_{n\to\infty}a_n^2=a^2$ when ${a_n}$ is a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$
negative_passages.docid: 47606
negative_passages.text: Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
negative_passages.docid: 91901
negative_passages.text: Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$
negative_passages.docid: 44507
negative_passages.text: How can I find $\lim_{n\to \infty} a_n$
negative_passages.docid: 119200
negative_passages.text: When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail
negative_passages.docid: 122820
negative_passages.text: Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
negative_passages.docid: 64329
negative_passages.text: Limit of the recursive sequence $a_{n+1} = \frac{1}{2}(a_n + \frac{p}{a_n})$
negative_passages.docid: 31174
negative_passages.text: Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
negative_passages.docid: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 17090
negative_passages.text: Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$
negative_passages.docid: 27410
negative_passages.text: If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
negative_passages.docid: 27932
negative_passages.text: Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$
negative_passages.docid: 64932
negative_passages.text: Show that $\lim_{x\to a^{+}} g(x) =g(a)$