query: $a_1=1$, $a_{n+1}=\frac{1}{2}\left(a_n+\frac{2}{a_n}\right)$. Show that the sequence is decreasing
positive_passages.docid: 152129
positive_passages.text: $a_1=3$ and $a_{n+1}=\frac{a_n}{2} + \frac{1}{a_n}$. Show that it monotonically decreases and find the limit.
negative_passages.docid: 49497
negative_passages.text: $\lim a_n = 0$ if $\left|\frac{a_{n +1}}{a_n}\right|\to L < 1$
negative_passages.docid: 95862
negative_passages.text: Prove that $t_{n-1, \alpha/2}$ is strictly decreasing in $n$
negative_passages.docid: 35611
negative_passages.text: How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
negative_passages.docid: 123253
negative_passages.text: Limit of a sequence $a_1=1;a_{n+1}=(n+1)(1+a_n)$
negative_passages.docid: 30471
negative_passages.text: Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges.
negative_passages.docid: 128128
negative_passages.text: Show that $\lim_{n \to \infty} \left(1+\frac{a_n}{n}\right)^n= e^a$ when $a_n \to a$
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: 73892
negative_passages.text: Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$
negative_passages.docid: 25583
negative_passages.text: Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing
negative_passages.docid: 9044
negative_passages.text: Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$
negative_passages.docid: 60405
negative_passages.text: Showing a recursive sequence isn't bounded $a_{n+1}=a_n+\frac 1 {a_n}$
negative_passages.docid: 13390
negative_passages.text: Recursive Monotone Decreasing Sequence Proof ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$
negative_passages.docid: 112663
negative_passages.text: Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity
negative_passages.docid: 61670
negative_passages.text: Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$
negative_passages.docid: 120874
negative_passages.text: How to show if $|a_{n+1} - a_{n}| \le \frac{1}{2^n}$ then the sequence is Cauchy.
negative_passages.docid: 74522
negative_passages.text: Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$