query: Show that $\left( 1+\dfrac {1} {n}\right) ^{n}$ is increasing
positive_passages.docid: 25921
positive_passages.text: Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
negative_passages.docid: 128128
negative_passages.text: Show that $\lim_{n \to \infty} \left(1+\frac{a_n}{n}\right)^n= e^a$ when $a_n \to a$
negative_passages.docid: 121774
negative_passages.text: Prove that $ \left(a_{n}\right)_{n=1}^{\infty} $ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $ 0<q<1 $
negative_passages.docid: 112663
negative_passages.text: Let $a_n$ be a sequence s.t $a_1 > 0 \land a_{n+1} = a_n + \dfrac{1}{a_n}$. Prove that $a_n$ is increasing and tends to infinity
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 14951
negative_passages.text: Other Idea to show an inequality $\dfrac{1}{\sqrt 1}+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\cdots+\dfrac{1}{\sqrt n}\geq \sqrt n$
negative_passages.docid: 73865
negative_passages.text: Find $\lim_{n\to\infty}n\left(\sum_{i=1}^{n}\dfrac{1}{(n+i)^2}\right)$
negative_passages.docid: 92175
negative_passages.text: Given $y_n=(1+\frac{1}{n})^{n+1}$ show that $\lbrace y_n \rbrace$ is a decreasing sequence
negative_passages.docid: 39270
negative_passages.text: Showing that $e^x > 1+x$
negative_passages.docid: 98768
negative_passages.text: Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
negative_passages.docid: 26962
negative_passages.text: Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$
negative_passages.docid: 34017
negative_passages.text: Show that $n^{\pi\left(2n\right)-\pi\left(n\right)}2$
negative_passages.docid: 35611
negative_passages.text: How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
negative_passages.docid: 71639
negative_passages.text: Proof $(1+1/n)^n$ is an increasing sequence
negative_passages.docid: 77510
negative_passages.text: $\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$
negative_passages.docid: 24683
negative_passages.text: For i.i.d. $\{X_n\}$ proving $\frac1n\max\limits_{1\leqslant k\leqslant n}|X_k|\xrightarrow{\mathrm{a.s.}}0\Leftrightarrow E(|X_1|)<+∞$