query: Show that $|\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k}{n})-\int_0^1f(x)dx|\leq\frac{M}{2n}$
positive_passages.docid: 20431
positive_passages.text: Speed of convergence of Riemann sums
negative_passages.docid: 51837
negative_passages.text: Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
negative_passages.docid: 9046
negative_passages.text: Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$
negative_passages.docid: 119133
negative_passages.text: there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$
negative_passages.docid: 15441
negative_passages.text: Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
negative_passages.docid: 5272
negative_passages.text: Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
negative_passages.docid: 80553
negative_passages.text: Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$
negative_passages.docid: 22864
negative_passages.text: Show that $\int_{a}^{b} |f|^2\le\frac{(b-a)^2}{2}\int_{a}^{b} |f'|^2$
negative_passages.docid: 50972
negative_passages.text: Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
negative_passages.docid: 102649
negative_passages.text: Prove that $\int_0^1\frac{x\ln (1+x)}{1+x^2}dx=\frac{\pi^2}{96}+\frac{\ln^2 2}{8}$
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 158180
negative_passages.text: An inequality of $\int_0^1 |f(x)|dx$
negative_passages.docid: 2414
negative_passages.text: If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
negative_passages.docid: 80826
negative_passages.text: To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
negative_passages.docid: 33097
negative_passages.text: Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$
negative_passages.docid: 118559
negative_passages.text: Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
negative_passages.docid: 124654
negative_passages.text: Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac1{2^n - 1} \le n$