query: If $\int_0^1 f(x)dx$ exist but $\lim_{x\to 0}f(x)$ doesn't exist, does $\lim_{n\to \infty }\frac{1}{n}\sum_{k=1}^n f(k/n)=\int_0^1 f(x)dx$?
positive_passages.docid: 143240
positive_passages.text: Convergence of Riemann sums for improper integrals
negative_passages.docid: 133405
negative_passages.text: $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?
negative_passages.docid: 6620
negative_passages.text: If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
negative_passages.docid: 102581
negative_passages.text: Does $\sum_{n=1}^{\infty} \frac{1}{n^{1 + 1/n}}$ converge?
negative_passages.docid: 41515
negative_passages.text: fine the limit : $\lim_{ n \to \infty }\frac{1}{n}\int_{0}^{n}{ \frac{x\ln(1+\frac{x}{n})}{1+x}}=?$
negative_passages.docid: 119200
negative_passages.text: When does $\lim_{n\to\infty}f(x+\frac{1}{n})=f(x)$ a.e. fail
negative_passages.docid: 104853
negative_passages.text: Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
negative_passages.docid: 155475
negative_passages.text: $\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent
negative_passages.docid: 71798
negative_passages.text: $\lim_{n\to \infty}\frac{1}{n} \sum_{k=1}^n x_k =x\;$ given $\;\lim_{n\to \infty} x_n= x\;?$
negative_passages.docid: 134787
negative_passages.text: How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
negative_passages.docid: 62504
negative_passages.text: Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$
negative_passages.docid: 147887
negative_passages.text: When do we have $ \lim_{n\to\infty}\sum_{k=1}^{n^2}\frac{n^{r-1}}{n^r+k^r}0$?
negative_passages.docid: 107386
negative_passages.text: Compute $\lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^n (\sin{x})^kdx$
negative_passages.docid: 71071
negative_passages.text: Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
negative_passages.docid: 80277
negative_passages.text: How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
negative_passages.docid: 150305
negative_passages.text: $f_{n+1}(x):= \int_a ^x f_n(t)dt$, $\sum_{m=1} ^{\infty} f_m(x)$ is uniformly convergent
negative_passages.docid: 65993
negative_passages.text: Does $\sum_{k=1}^\infty\frac1{k^n}$ converge for $\Re(n)=1,\Im(n)\ne0$?