query: Find $\ \limsup_{n\to \infty}(\frac{2^n}{n!}) $
positive_passages.docid: 2449
positive_passages.text: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
negative_passages.docid: 146663
negative_passages.text: Limit $\lim_{n \to \infty} \frac {a_n}{2^{n-1}}=\frac 4{\pi}$ for $a_{n+1}=a_n+\sqrt{1+a_n^2}$ and $a_0=0$
negative_passages.docid: 81561
negative_passages.text: Limit of factorial function: $\lim\limits_{n\to\infty}\frac{n^n}{n!}.$
negative_passages.docid: 24941
negative_passages.text: How to evaluate $\lim_{n\to\infty}n!/n^{k}$
negative_passages.docid: 18062
negative_passages.text: How can I prove $\lim_{n \to \infty} \frac{n^2}{2^n}=0$
negative_passages.docid: 153269
negative_passages.text: Find $\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$
negative_passages.docid: 52331
negative_passages.text: What is $\lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right)$?
negative_passages.docid: 113289
negative_passages.text: finding the limit $\lim\limits_{x \to \infty }(\frac{1}{e}(1+\frac{1}{x})^x)^x$
negative_passages.docid: 111636
negative_passages.text: Finding $\lim_{n\to\infty}\int_0^{\frac{1}{2^n}}(1+\cos(x))^n\,dx$
negative_passages.docid: 93095
negative_passages.text: Find $\lim\limits_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)$
negative_passages.docid: 83634
negative_passages.text: Find $\lim_{n\to \infty}\sqrt[n]{3^n+5^n}$
negative_passages.docid: 61749
negative_passages.text: Limit of the infinite sum of $\frac{n}{2^n}$?
negative_passages.docid: 56961
negative_passages.text: Evaluate the limit of $\sum_{n=1}^{\infty}\frac{n^2}{n!}$
negative_passages.docid: 152713
negative_passages.text: Using the squeeze theorem to determine a limit $\lim_{n\to\infty} (n!)^{\frac{1}{n^2}}$
negative_passages.docid: 128938
negative_passages.text: Finding $\lim_{n\to\infty} a_n^n$ where $a_n=\frac{(n!)^2}{(n+k)!(n-k)!}$
negative_passages.docid: 19631
negative_passages.text: Finding $\lim_{n\to \infty}\sqrt n \int_0^1 \frac{\,dx}{(1+x^2)^n}$
negative_passages.docid: 96848
negative_passages.text: Prove: $\limsup_{n\to\infty} (a_n)\leq \sup(a_n)_{n=1}^\infty$