query: Proove that the sequence $\left (1+\frac{1}{x} \right )^{x}$ increases.
positive_passages.docid: 25775
positive_passages.text: How to prove $(1+1/x)^x$ is increasing when $x>0$?
negative_passages.docid: 16392
negative_passages.text: Prove the Inequality on sequence
negative_passages.docid: 6100
negative_passages.text: Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
negative_passages.docid: 27717
negative_passages.text: Compute $\lim_{x\to\infty}x\;\left[\left(1+\frac{1}{x}\right)^x-e\right]$
negative_passages.docid: 91528
negative_passages.text: sequence $x_{n+1}=x_n+e^{-2018x_n}$
negative_passages.docid: 17779
negative_passages.text: Proving the Limit of a Sequence of without the Dominated Convergence Theorem
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: 94409
negative_passages.text: showing that the sequence $a_n=1+\frac{1}{2}+...+\frac{1}{n} - \log(n)$ converges
negative_passages.docid: 21589
negative_passages.text: Prove the divergence of the sequence $\left\{ \sin(n) \right\}_{n=1}^{\infty}$.
negative_passages.docid: 71547
negative_passages.text: Prove $ 1+\left( 1-\frac{x}{n}\right)+...+\left(1-\frac{x}{n}\right)^{n-1}=\frac{n}{x}\left(1-\left(1-\frac{x}{n}\right)^n\right)$
negative_passages.docid: 13125
negative_passages.text: Convergence of $\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$
negative_passages.docid: 102523
negative_passages.text: Show that $\lim\limits_{n \to \infty} \sup\limits_{k \geq n} \left(\frac{1+a_{k+1}}{a_k}\right)^k \ge e$ for any positive sequence $\{a_n\}$
negative_passages.docid: 125497
negative_passages.text: Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$
negative_passages.docid: 138516
negative_passages.text: $\sum (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$ convergent but not absolutely convergent
negative_passages.docid: 100580
negative_passages.text: Proof that a sequence converges.
negative_passages.docid: 131847
negative_passages.text: $\frac{(x-1)(x+1)}{ (x+1)} \rightarrow (x-1)$ Domain Change
negative_passages.docid: 112595
negative_passages.text: Prove that the limit of this sequence is $\frac{1}{2}(1 + \sqrt{5})$