query: Limit of S_n, where $S_n = \int_{0}^{1}\frac{nx^{n-1}}{1+x} dx$ for $n\ge 1$
positive_passages.docid: 11489
positive_passages.text: Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$
negative_passages.docid: 9541
negative_passages.text: Evaluate $\lim_{n\to\infty}\frac {qn+1}{qn} \frac {qn+p+1}{qn+p} \cdots \frac {qn+np+1}{qn+np}$ where $p\in\mathbb{N}\setminus\{0,1\},q>0.$
negative_passages.docid: 155513
negative_passages.text: I need help to advance in the resolution of that limit: $ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $
negative_passages.docid: 106087
negative_passages.text: Finding $\lim_{n \to \infty} \int_1^a \frac{n}{1+x^n} \, dx$
negative_passages.docid: 65137
negative_passages.text: Evaluating $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ where $a_1 \ge \cdots\ge a_k \ge 0$
negative_passages.docid: 66526
negative_passages.text: Find the limit of: $\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$
negative_passages.docid: 117640
negative_passages.text: Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
negative_passages.docid: 17157
negative_passages.text: Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
negative_passages.docid: 111279
negative_passages.text: The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$
negative_passages.docid: 27748
negative_passages.text: Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$
negative_passages.docid: 49128
negative_passages.text: How to prove $\lim\limits_{n \to \infty} (1+\frac1n)^n = e$?
negative_passages.docid: 105860
negative_passages.text: Evaluate the limit of $(n+1)\int_0^1x^n\ln(1+x)\,dx$ when $n\to\infty$
negative_passages.docid: 103513
negative_passages.text: Show that $\lim_{n\to \infty}\int_{-\pi}^{\pi}f(x)g(nx)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\int_{-\pi}^{\pi}g(x)\,dx$
negative_passages.docid: 68368
negative_passages.text: Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$
negative_passages.docid: 15822
negative_passages.text: Value of $\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$
negative_passages.docid: 34025
negative_passages.text: Limit of the Sequence: $a_{n+1}=a_n+\frac{1}{a_n}$ with $a_1=1$
negative_passages.docid: 95095
negative_passages.text: $x_{n+m}\le \frac{x_n+x_{n+1}+\cdots+x_{n+m-1}}{m}$. Prove that this sequence has a limit.