query: is the sequence $\{a_n\}$ is strictly increasing or strictly decreasing?
positive_passages.docid: 25294
positive_passages.text: Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$
negative_passages.docid: 75317
negative_passages.text: Show that $(s_n)$ is a Cauchy sequence
negative_passages.docid: 49419
negative_passages.text: Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?
negative_passages.docid: 108696
negative_passages.text: If $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence, then $(a_n ^2)_{n\in\mathbb{N}}$ is also a Cauchy sequence
negative_passages.docid: 47922
negative_passages.text: Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$
negative_passages.docid: 86319
negative_passages.text: Give an example of a bounded, non-convergent real sequence $(a_n)$ s.t. $a_n-a_{n-1}\rightarrow 0$
negative_passages.docid: 92175
negative_passages.text: Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence
negative_passages.docid: 132358
negative_passages.text: Is the sequence $a_{n+1}=a_n-\frac{1}{a_n}$, $a_0=2$ bounded?
negative_passages.docid: 95860
negative_passages.text: The limit of the difference of two consecutive sequence members is equal to $0$. Can we conclude that the sequence itself has a limit?
negative_passages.docid: 51555
negative_passages.text: Strictly increasing sequence $\{ a_k \}$ of positive integers such that $\sum 1/{a_k}$ is finite
negative_passages.docid: 73553
negative_passages.text: How is $a_n=(1+1/n)^n$ monotonically increasing and bounded by $3$?
negative_passages.docid: 100580
negative_passages.text: Proof that a sequence converges.
negative_passages.docid: 95845
negative_passages.text: Is there a sequence such that $\sum{a_n}$ diverges but $\sum{na_n}$converges?
negative_passages.docid: 135018
negative_passages.text: Proving that a sequence has $a_n = a_{n+2}$ for $n$ sufficiently large.
negative_passages.docid: 151226
negative_passages.text: Why is the sequence $x(n) = \log n$ **not** Cauchy?
negative_passages.docid: 2092
negative_passages.text: Convergence of a sequence with assumption that exponential subsequences converge?
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.