query: $\langle x_n\rangle $ be a sequence such that $x_{(n+2)}=\frac{(x_{n+1}+ x_{n})}{2} $for all $n \in \mathbb{N}$, prove that $x_n$ is convergent
positive_passages.docid: 142509
positive_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: 104344
negative_passages.text: Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent
negative_passages.docid: 54659
negative_passages.text: How prove that $(a_{n})_{n \in \mathbb{N}}$ is convergent with $a_{0}=1$, $a_{n+1}= \sqrt{2a_{n}}$?
negative_passages.docid: 90936
negative_passages.text: how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?
negative_passages.docid: 99933
negative_passages.text: Determine whether $(x_n)$ converges if $x_1= a , x_{n+1} = x_n+\frac{1}{x_n}$ for $n \in \mathbb{N}$
negative_passages.docid: 67638
negative_passages.text: Show that $x_{n+1}=x_n(2-ax_n)$ converges and find the limit
negative_passages.docid: 54982
negative_passages.text: How to prove that $x_n=1+\frac{1}{2!}+\cdots+\frac{1}{n!}$ is a Cauchy sequence?
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: 83040
negative_passages.text: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
negative_passages.docid: 60024
negative_passages.text: Proof that a sequence is convergent
negative_passages.docid: 153121
negative_passages.text: show that $\{x_n\}$ is a cauchy sequence
negative_passages.docid: 107399
negative_passages.text: A sequence and convergence
negative_passages.docid: 55914
negative_passages.text: If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent
negative_passages.docid: 79266
negative_passages.text: Proving a sequence defined by a recurrence relation converges
negative_passages.docid: 147841
negative_passages.text: Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$
negative_passages.docid: 49507
negative_passages.text: Is this sequence convergent?
negative_passages.docid: 87834
negative_passages.text: How can we show that the following sequence converges?