query: With a sequence $a_n$ that satisfies $\lim\limits_{n\to\infty}(a_{n} - a_{n-2})=0$, prove or disprove this statement
positive_passages.docid: 27410
positive_passages.text: If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
negative_passages.docid: 98167
negative_passages.text: If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$
negative_passages.docid: 5797
negative_passages.text: Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$
negative_passages.docid: 144165
negative_passages.text: $n\in N, a_{n+1} = a_n (1 - a_n)$ , $0 < a_0 < 1$. Prove that $\lim\limits_{n\to\infty} a_n\cdot n = 1$.
negative_passages.docid: 142778
negative_passages.text: Finding $\lim_{n\to\infty}a_n$ if $a_0=a,a_1=b$ and $a_n=\frac{1}{2}(a_{n-1}+a_{n-2})$ where $a, b \in \mathbb{R}$
negative_passages.docid: 145863
negative_passages.text: sequence of positive numbers satisfying $a_{n+1}=\frac{2}{a_n+a_{n-1}}$, prove it converges
negative_passages.docid: 53688
negative_passages.text: How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
negative_passages.docid: 155695
negative_passages.text: The convergence of a sequence
negative_passages.docid: 139356
negative_passages.text: How to find the limit of a sequence?
negative_passages.docid: 6878
negative_passages.text: Prove limit of sequence
negative_passages.docid: 67645
negative_passages.text: Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if $a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$
negative_passages.docid: 119776
negative_passages.text: Prove or disprove: if $\sum\limits_{n=0}^∞a_n$ converges and $\lim\limits_{n\to\infty}b_n =0$, then $\sum\limits_{n=0}^{\infty}a_n b_n$ converges.
negative_passages.docid: 112388
negative_passages.text: Suppose $\lim \limits_{n \to ∞} a_n=L$. Prove that $\lim\limits_{n \to ∞} \frac{a_1+a_2+\cdots+a_n}{n}=L$
negative_passages.docid: 116290
negative_passages.text: How to give a example that $\lim\limits_{n\to\infty}n|a_{n}-a_{n+1}|=0$ and {$a_{n}$} divergent
negative_passages.docid: 149068
negative_passages.text: Prove: Assume $\lim_{n\to\infty}(a_n)=a, \lim_{n\to\infty}(b_n)=b$ then $\lim_{n\to\infty}(a_n+b_n)=a+b$
negative_passages.docid: 111211
negative_passages.text: Does $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}=1$ imply that $a_n$ convergent?
negative_passages.docid: 26837
negative_passages.text: Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?