query: $\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{k}{k^2 + 1}$
positive_passages.docid: 90773
positive_passages.text: $\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{k}{k^{2}+1}$
negative_passages.docid: 67500
negative_passages.text: Sum of $\lim_{n\rightarrow \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+2}+\cdots \cdots \cdots +\frac{n}{n^2+n}\right)$
negative_passages.docid: 62443
negative_passages.text: Prove that $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ converges and find its sum
negative_passages.docid: 8489
negative_passages.text: How to compute $\lim_{n\rightarrow\infty}\frac1n\left\{(2n+1)(2n+2)\cdots(2n+n)\right\}^{1/n}$
negative_passages.docid: 64223
negative_passages.text: Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
negative_passages.docid: 5433
negative_passages.text: Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
negative_passages.docid: 29901
negative_passages.text: Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$.
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: 11389
negative_passages.text: What is $\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$?
negative_passages.docid: 41494
negative_passages.text: Find limit $\lim_{n\to \infty}\frac{\sum_{k=1}^n k^n}{n^n}$.
negative_passages.docid: 95718
negative_passages.text: Computing $ \lim_{n\rightarrow \infty}\left(\frac{1^n+2^n+\cdots \cdots +n^n}{n^n}\right)$
negative_passages.docid: 59339
negative_passages.text: How to prove that $ \lim_{n\to\infty}\left( \sum_{r=1}^n \dfrac 1 {\sqrt{n^2 + r}} \right) = 1$
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: 75273
negative_passages.text: Help needed with definite integral for $\lim_{n\rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n+k}$
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: 82831
negative_passages.text: Evaluating the limit $\lim_{n\to\infty} \left[ \frac{n}{n^2+1}+ \frac{n}{n^2+2^2} + \ldots + \frac{n}{n^2+n^2} \right]$
negative_passages.docid: 106281
negative_passages.text: Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$