query: $|\sin(nx)| ≤ n|\sin x|$ Proof by induction
positive_passages.docid: 16253
positive_passages.text: Proof that $|\sin(nx)| \le n|\sin(x)|$
negative_passages.docid: 30783
negative_passages.text: Prove inequality $n<3^n$ using mathematical induction
negative_passages.docid: 90556
negative_passages.text: Proof of the unbounded-ness of $\sum_{n\geq 1}\frac{1}{n}\sin\frac{x}{n}$
negative_passages.docid: 146180
negative_passages.text: Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
negative_passages.docid: 137844
negative_passages.text: Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$
negative_passages.docid: 139269
negative_passages.text: Prove that $2n-3\leq 2^{n-2}$ , for all $n \geq 5$ by mathematical induction
negative_passages.docid: 52115
negative_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: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 108382
negative_passages.text: Prove $x^n-1$ is divisible by $x-1$ by induction
negative_passages.docid: 138036
negative_passages.text: Proving Inequality using Induction $a^n-b^n \leq na^{n-1}(a-b)$
negative_passages.docid: 32823
negative_passages.text: Prove $\forall n \in \mathbb{N}: \int_{0}^{\frac{\pi}{2}} |\frac{\sin(nx)}{x}|dx \geq \frac{2}{\pi}\sum_{k=1}^{n}\frac{1}{k}$
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 128795
negative_passages.text: $n$th derivative of $e^x \sin x$
negative_passages.docid: 40420
negative_passages.text: Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
negative_passages.docid: 41784
negative_passages.text: Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
negative_passages.docid: 95085
negative_passages.text: Prove by induction $ \sin x + \sin 2x + ... + \sin nx = \frac {\sin (\frac {n + 1} {2} x)} {\sin \frac{x}{2}} \sin \frac{nx}{2} $
negative_passages.docid: 95030
negative_passages.text: Proving the Cauchy-Schwarz inequality by induction