query: $\lim_{|P|\to0} \sum_{i=1}^n f(\xi_i)g(\eta_i)(t_{i}-t_{i-1})=\int_a^b f(x)g(x)dx$
positive_passages.docid: 79590
positive_passages.text: Riemann and Darboux Integral of a product of two functions
negative_passages.docid: 8913
negative_passages.text: Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
negative_passages.docid: 154612
negative_passages.text: $\int_{0}^{x} f(t) dt \ge {f(x)}$ holds or not on $[0,1]$
negative_passages.docid: 10408
negative_passages.text: Uniformly convergence of $f_{n+1}(x)= \int^x_0 f_n(t)dt$
negative_passages.docid: 35325
negative_passages.text: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
negative_passages.docid: 151568
negative_passages.text: Limit Integral equal to a supremun: $\lim_{n\to \infty} \left[\int_a^b (f(x))^n dx\right]^{{1}/{n}}=\sup_{x\in [a,b]} f(x).$
negative_passages.docid: 93806
negative_passages.text: $\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
negative_passages.docid: 15045
negative_passages.text: Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$
negative_passages.docid: 64932
negative_passages.text: Show that $\lim_{x\to a^{+}} g(x) =g(a)$
negative_passages.docid: 64571
negative_passages.text: Prove that $\int_{E}f =\lim \int_{E}f_{n}$
negative_passages.docid: 70304
negative_passages.text: Prove there are $\xi$, $\eta$ with $f'(\xi)f'(\eta)=1$
negative_passages.docid: 138169
negative_passages.text: Finding $\lim\limits_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$
negative_passages.docid: 97266
negative_passages.text: Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$
negative_passages.docid: 15262
negative_passages.text: Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
negative_passages.docid: 152557
negative_passages.text: Calculate $ \lim\limits_{n \to \infty} \sum\limits_{k=0}^n \frac{2^{k-1}}{a^{2k-1}+1} $
negative_passages.docid: 117246
negative_passages.text: Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$.
negative_passages.docid: 42519
negative_passages.text: $\lim_{n \rightarrow \infty} \sum^n_{k=1} \frac{1}{k+n}$