query: $\lim x_n = a \in \mathbb{R}$, $\lim y_n = b \in \mathbb{R}$. Prove that $\lim \frac{x_1y_n + x_2y_{n-1} + \cdots + x_ny_1}{n} = ab$
positive_passages.docid: 78393
positive_passages.text: Find the limit $\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}$
negative_passages.docid: 52819
negative_passages.text: limit laws:$\lim_{n\to\infty}\max(a_n,b_n)=\max(\lim_{n\to\infty}a_n,\lim_{n\to\infty}b_n)$
negative_passages.docid: 90377
negative_passages.text: Proving the general limit $\lim_{n\to\infty} \frac{an+c}{bn+d} = \frac{a}{b}$
negative_passages.docid: 122820
negative_passages.text: Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
negative_passages.docid: 11879
negative_passages.text: If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
negative_passages.docid: 13249
negative_passages.text: Multivariable limit proof: $\lim\limits_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$
negative_passages.docid: 57418
negative_passages.text: How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$
negative_passages.docid: 17289
negative_passages.text: prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit
negative_passages.docid: 100733
negative_passages.text: $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$
negative_passages.docid: 53918
negative_passages.text: whats the proof for $\lim_{x → 0} [(a_1^x + a_2^x + .....+ a_n^x)/n]^{1/x} = (a_1.a_2....a_n)^{1/n}$
negative_passages.docid: 29287
negative_passages.text: Prove that: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
negative_passages.docid: 109576
negative_passages.text: Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
negative_passages.docid: 50673
negative_passages.text: Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
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: 36060
negative_passages.text: If $\lim_{n\to\infty}x_n=\alpha$, show $\lim_{n\to\infty}(x_1x_2\dots x_n)^{1/n} =\alpha$.
negative_passages.docid: 22347
negative_passages.text: Proving that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$
negative_passages.docid: 91901
negative_passages.text: Prove that $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$