query: Prove: If $|r| < 1$, then $\lim_{n\to \infty }r^n$ = 0.
positive_passages.docid: 1760
positive_passages.text: Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$?
negative_passages.docid: 153267
negative_passages.text: Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
negative_passages.docid: 35325
negative_passages.text: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
negative_passages.docid: 130161
negative_passages.text: Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$
negative_passages.docid: 108917
negative_passages.text: Use $\delta-\epsilon$ to show that $\lim_{n\to\infty} a^{\frac{1}{n}} = 1$?
negative_passages.docid: 9694
negative_passages.text: If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists
negative_passages.docid: 72693
negative_passages.text: Prove: $\lim_{x\to\infty}x^{2/3}((x+1)^{1/3}-x^{1/3})=1/3$
negative_passages.docid: 158153
negative_passages.text: Prove: $ \lim_{n\rightarrow\infty} x ^ {\left(\frac 1 n\right)} = 1 $
negative_passages.docid: 48593
negative_passages.text: Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
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: 80373
negative_passages.text: How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$
negative_passages.docid: 116966
negative_passages.text: Prove $\limsup_{n \rightarrow \infty} a_n\geq1$ if $\lim_{n \rightarrow \infty} a_na_{n+1}=1$
negative_passages.docid: 119804
negative_passages.text: If $\lim_{n\to \infty}a_n = a\in \mathbb{R}$ . Prove that $\limsup_{n\to \infty}a_n x_n=a\limsup_{n\to \infty}x_n$ .
negative_passages.docid: 7815
negative_passages.text: Prove that $\lim_{n\to\infty}a_n^2=a^2$ when ${a_n}$ is a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$
negative_passages.docid: 67084
negative_passages.text: Prove or disprove that $\lim_{n \to \infty} {n \choose k} / a^n = 0$ for all $k \in \mathbb{Z_+}, a \in \mathbb{R}: a> 1$.
negative_passages.docid: 73812
negative_passages.text: Does $\lim \limits_ {n\to \infty}\ \frac{n}{n+1}$ equal $1$?
negative_passages.docid: 152223
negative_passages.text: If $0<r<1$ show that $r^n$ goes to $0$ as $n$ goes to 1.