query: Trying to prove that $\lim_{N \rightarrow \infty} \frac{1}{N} \Sigma_{n=1}^N f(n\alpha) = \int_0^1 f(x) dx$
positive_passages.docid: 20089
positive_passages.text: Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
negative_passages.docid: 14129
negative_passages.text: How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
negative_passages.docid: 36060
negative_passages.text: If $\lim_{n\to\infty}x_n=\alpha$, show $\lim_{n\to\infty}(x_1x_2\dots x_n)^{1/n} =\alpha$.
negative_passages.docid: 110588
negative_passages.text: Showing $\lim_{n\rightarrow\infty}\sqrt[3]{n^3+n^2}-\sqrt[3]{n^3+1}\rightarrow\frac{1}{3}$
negative_passages.docid: 25206
negative_passages.text: How to prove that $\lim\limits_{n\to \infty}a_{n+1}=\lim\limits_{n\to \infty}a_{n}$
negative_passages.docid: 87116
negative_passages.text: Prove $ \lim_{n\to \infty}\frac{n}{n+1} = 1$ using Cauchy (epsilon) definition of a limit of a sequence
negative_passages.docid: 143227
negative_passages.text: Show that: $\lim \limits_{n\to\infty}\frac{x_n-x_{n-1}}{n}=0 $
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: 116257
negative_passages.text: How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$
negative_passages.docid: 42817
negative_passages.text: proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]
negative_passages.docid: 11879
negative_passages.text: If $\lim_{n\rightarrow\infty} f(n+1) - f(n) = L$, prove that $\lim_{n\rightarrow\infty} f(n)/n = L$
negative_passages.docid: 1371
negative_passages.text: Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
negative_passages.docid: 137609
negative_passages.text: Proving $\lim_{n\rightarrow\infty} n(2^{1/n} - 1) = \log 2$
negative_passages.docid: 20956
negative_passages.text: Correct way to prove the limit: $\mathop {\lim }\limits_{x \to 0} {x^\alpha }\int\limits_x^1 {\frac{{f\left( t \right)}}{{{t^{\alpha  + 1}}}}dt} $?
negative_passages.docid: 104996
negative_passages.text: Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$
negative_passages.docid: 138973
negative_passages.text: How prove this limit $\lim\limits_{n\rightarrow \infty} \frac{f_n}{f_{n+1}}=a$ given two other limits related to $f_n$
negative_passages.docid: 75268
negative_passages.text: Proving that $\lim_{n\rightarrow \infty} \frac{n^k}{2^n}=0$