query: If $(x_{2k+4})$ and $(x_{2k+5})$ converge to the same limit, is $(x_k)$ also converging, and to the same limit?
positive_passages.docid: 74542
positive_passages.text: Convergence of a sequence whose even and odd subsequences converge
negative_passages.docid: 49929
negative_passages.text: If $ \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it true that $ \lim_{n \to \infty} x_{n} = x $?
negative_passages.docid: 27284
negative_passages.text: Show that the sequence $\sqrt{2}, \ \sqrt{2+\sqrt{2}}, \ \sqrt{2+\sqrt{2+\sqrt{2+}}}...$ converges and find its limit.
negative_passages.docid: 97968
negative_passages.text: Does $\big|\sum_{k=1}^na_k\big|\leq\sqrt{n}$ imply the convergence of $\sum_{1}^\infty \frac{a_k}{k}$?
negative_passages.docid: 14680
negative_passages.text: $\prod_{k=1}^n (1+ \frac{z}{k})$ converges to $0$ when $\Re (z)<0$
negative_passages.docid: 142509
negative_passages.text: If $\alpha\in (0,1]$ and $x_n=\alpha x_{n-1}+(1-\alpha)x_{n-2}$, show that the sequence $\{x_n\}$ is convergent.
negative_passages.docid: 95515
negative_passages.text: Is $x_{n+1}=\cos x_n$ convergent?
negative_passages.docid: 84969
negative_passages.text: When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges?
negative_passages.docid: 35885
negative_passages.text: Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$
negative_passages.docid: 155695
negative_passages.text: The convergence of a sequence
negative_passages.docid: 147849
negative_passages.text: Does $(n!)/n^{1000}$ diverge or converge
negative_passages.docid: 19225
negative_passages.text: Let $\{a_n\}$ be defined as follows: $a_1 = 2$, $a_{n+1}=\frac{1}{3-a_n}$, if $n \geq 1$. Does $\{a_n\}$ converge?
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: 14243
negative_passages.text: Is it possible that $a_n>0$ and $\sum a_n$ converges then $na_n \to 0$? (without assuming $a_n$ is decreasing)
negative_passages.docid: 10208
negative_passages.text: Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$
negative_passages.docid: 69989
negative_passages.text: Do the following series converge if $a_n>0$ and $ \sum_{n=1}^{\infty}a_n$ diverges?
negative_passages.docid: 32423
negative_passages.text: Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$