query: Prove that $1+x+\frac{x^2}{2}+\dots+\frac{x^n}{n!}<e^x$ for all $x\in(0,\infty),n\in\mathbb{N}$
positive_passages.docid: 52115
positive_passages.text: Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction
negative_passages.docid: 108405
negative_passages.text: Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$
negative_passages.docid: 38062
negative_passages.text: Proving the convergence of $\sum\limits_{n=1}^{\infty}\frac{1}{1+z^n}$ for $|z| > 1$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 51653
negative_passages.text: Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
negative_passages.docid: 66902
negative_passages.text: Prove that $\int_{-\infty}^{\infty} \frac{e^{2x}-e^x}{x (1+e^{2x})(1+e^{x})}\mathrm{d}x=\log 2$
negative_passages.docid: 150565
negative_passages.text: Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
negative_passages.docid: 15292
negative_passages.text: Elementary proof for $\lim\limits_{n \to\infty}\frac{n!e^n}{n^n} = +\infty$
negative_passages.docid: 72748
negative_passages.text: Proof that $\sum\limits^{\infty}_{i=0}{iP(X > i)} = \frac12(E(X²)-E(X))$ for $X$ nonnegative and integer valued
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: 128285
negative_passages.text: Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?
negative_passages.docid: 32407
negative_passages.text: Prove $\exp(x+y) = \exp(x) \exp(y)$ for $\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 142360
negative_passages.text: How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
negative_passages.docid: 21941
negative_passages.text: Proving: $x = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$
negative_passages.docid: 80216
negative_passages.text: Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
negative_passages.docid: 115912
negative_passages.text: Summation inductional proof: $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<2$