query: How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$
positive_passages.docid: 23119
positive_passages.text: An application of Jensen's Inequality
negative_passages.docid: 40790
negative_passages.text: Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.
negative_passages.docid: 93081
negative_passages.text: If $\sum\limits_{i=1}^na_i=\prod\limits_{i=1}^na_i$ for every $n$, identify $\lim\limits_{n\to \infty}a_n$
negative_passages.docid: 115514
negative_passages.text: Is the product $\prod_{k=1}^\infty \frac{2^k-1}{2^k}$ necessarily $0$?
negative_passages.docid: 30356
negative_passages.text: $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$...
negative_passages.docid: 14815
negative_passages.text: How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
negative_passages.docid: 15259
negative_passages.text: Proving $\prod \limits_{k=0}^{n}(1-a_k) \geq1- \sum\limits_{k=0}^{n}a_k$
negative_passages.docid: 91523
negative_passages.text: Prove this limit as $x\to\infty$
negative_passages.docid: 89523
negative_passages.text: Infinite product $(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})$
negative_passages.docid: 14450
negative_passages.text: Positive series problem: $\sum\limits_{n\geq1}a_n=+\infty$ implies $\sum_{n\geq1}\frac{a_n}{1+a_n}=+\infty$
negative_passages.docid: 103843
negative_passages.text: Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{\prod\limits_{k=1}^{m}(n+k)}$
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: 149022
negative_passages.text: In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$
negative_passages.docid: 13796
negative_passages.text: What is the value of $\prod_{n=1}^\infty (1+\frac{1}{n^2})$?
negative_passages.docid: 153905
negative_passages.text: If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $
negative_passages.docid: 21941
negative_passages.text: Proving: $x = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$