query: Convergence of sequence of means: Define $A_n=\frac{a_1+a_2+...a_n}{n}$, Find $\lim_{n\to\infty} \sqrt{n}(A_{n+1}-A_{n})$
positive_passages.docid: 46103
positive_passages.text: Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$
negative_passages.docid: 101766
negative_passages.text: Recursive sequence convergence.: $s_{n+1}=\frac{1}{2} (s_n+s_{n-1})$ for $n\geq 2$, where $s_1>s_2>0$
negative_passages.docid: 141463
negative_passages.text: Convergence of $x_{n+1} = \frac12(x_n + \frac2{x_n}).$
negative_passages.docid: 12480
negative_passages.text: Convergence of $\sum_{n=1}^{\infty} \frac{\frac{1}{2}+(-1)^n}{n}$
negative_passages.docid: 97047
negative_passages.text: Limit for sequence $a_{m+n}\leq a_m+a_n$
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: 19657
negative_passages.text: Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?
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: 157179
negative_passages.text: Cesaro Mean of Sequences - Convergence
negative_passages.docid: 44261
negative_passages.text: Convergence of $\sum^\infty_{n=1} \ln(1+\frac 1 {2^n})$
negative_passages.docid: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 26300
negative_passages.text: $a_n=(-1)^{n-1}, \; s_n=\sum_{i=1}^{n}a_i$ then find $ \lim_{n\to \infty}\frac{s_1+s_2+\dots s_n}{n}$
negative_passages.docid: 70561
negative_passages.text: limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ and $a_1=1$
negative_passages.docid: 74629
negative_passages.text: Find the limit of the sequence $\{\frac{x_1 + 2x_2 + ... + nx_n}{n^2}\}$, where $\{x_n\}$ is a convergent sequence in a normed space S.
negative_passages.docid: 93545
negative_passages.text: Showing the sequence converges $a_{1}=\frac{1}{2}$, $a_{n+1}=\frac{1}{2+a_{n}}$
negative_passages.docid: 47606
negative_passages.text: Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
negative_passages.docid: 91009
negative_passages.text: Convergence of the series $\sum a_n$ implies the convergence of $\sum \frac{\sqrt a_n}{n}$, if $a_n>0$