query: Show that $\lim \limits_{|P|\rightarrow 0} \sum \limits^n_{i=1}f(\zeta_i)g(\eta_i)(t_i-t_{i-1})=\int^{b}_{a}f(x)g(x)dx$
positive_passages.docid: 79590
positive_passages.text: Riemann and Darboux Integral of a product of two functions
negative_passages.docid: 60031
negative_passages.text: Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
negative_passages.docid: 21791
negative_passages.text: Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
negative_passages.docid: 90773
negative_passages.text: $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{k^{2}+1}$
negative_passages.docid: 142249
negative_passages.text: Show that $\lim_{n\to \infty} \int_E \cos^2(nx + a_n) dx = \frac{1}{2}m(E)$
negative_passages.docid: 157539
negative_passages.text: Prove: $\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=1}^{n}\sqrt{1+\frac{k}{n}}=\frac{2}{3}(2\sqrt{2}-1)$
negative_passages.docid: 150971
negative_passages.text: If $f$ is Riemann integrable on $[0,1]$ then $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum f\left(\frac{k}{n}\right)=\int\limits_{0}^{1} f(x)dx$
negative_passages.docid: 53213
negative_passages.text: How to prove that $\lim(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k}))=\chi_\mathbb{Q}$
negative_passages.docid: 80350
negative_passages.text: $\limsup \frac{a_n}{b_n}0$ s.t $a_n\le Mb_n$
negative_passages.docid: 17019
negative_passages.text: Show that: $\lim\limits_{r\to\infty}\int\limits_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$
negative_passages.docid: 132126
negative_passages.text: Proving $\sum\limits_{n=1}^{\infty } \frac{\mu (n)}n=0$ where $\mu$ is the Möbius function
negative_passages.docid: 86348
negative_passages.text: Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$
negative_passages.docid: 47606
negative_passages.text: Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit
negative_passages.docid: 56265
negative_passages.text: limit $\lim\limits_{n\to\infty}\left(\sum\limits_{i=1}^{n}\frac{1}{\sqrt{i}} - 2\sqrt{n}\right)$
negative_passages.docid: 11062
negative_passages.text: $f(z)=\frac1{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}d\zeta$ if $\lim_{z\to\infty}f(z)=0$ and $f$ is analytic in $|z|\ge1$
negative_passages.docid: 8626
negative_passages.text: The limit of a sum (a Riemann sum?)
negative_passages.docid: 104996
negative_passages.text: Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$