query: $\lim_{n\to\infty}(1+x)(1+x^{3})\cdots(1+x^{2n+1}),|x|<1.$
positive_passages.docid: 146221
positive_passages.text: Find the product $\prod_{n=1}^{\infty} (1+x^{2n-1}) $
negative_passages.docid: 61300
negative_passages.text: Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$
negative_passages.docid: 37277
negative_passages.text: $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
negative_passages.docid: 24695
negative_passages.text: How to solve the limit $\lim\limits_{n\to \infty}(1-a)(1-a^2)\cdots(1-a^n)$?
negative_passages.docid: 135700
negative_passages.text: Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
negative_passages.docid: 93095
negative_passages.text: Find $\lim\limits_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)$
negative_passages.docid: 78116
negative_passages.text: What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?
negative_passages.docid: 132988
negative_passages.text: $ \lim_{n \to \infty} \left(\frac 1{n^2+1}+\frac 2{n^2+2}+\frac 3{n^2+3}+\cdots +\frac n{n^2+n}\right)$
negative_passages.docid: 117370
negative_passages.text: $\lim_{n\to\infty} \left(1+\frac{2}{n}\right)^{n^2} e^{-2n}$
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: 25193
negative_passages.text: Find $ \lim _{n\to \infty} \frac{n^1+\dots+n^n}{1^n+\dots+n^n}$
negative_passages.docid: 17069
negative_passages.text: Limit $\lim_{x\to \infty} \frac{\sqrt{x+1}+\sqrt{x+2}-\sqrt{4x+12}}{\sqrt{x+2}+\sqrt{x}-\sqrt{4x+12}}$
negative_passages.docid: 41366
negative_passages.text: Compute $\lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{{1}/{2}}\cdots\left(1+\frac{n}{n}\right)^{{1}/{n}}$
negative_passages.docid: 151149
negative_passages.text: $\lim_{n \to \infty} n \int_{1}^{\infty} \frac{dx}{1+x^n}$, solution critique needed
negative_passages.docid: 147264
negative_passages.text: $ (1 + x)^{1/x} $ when $ x \to \infty$ without L’Hospital’s Rule
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: 66441
negative_passages.text: Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$