query: increasing numbers $x_n\to\infty$ such that $\sum_{n=1}^\infty(1-x_n/x_{n+1})<\infty$
positive_passages.docid: 149240
positive_passages.text: Divergence of a series
negative_passages.docid: 2678
negative_passages.text: Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$
negative_passages.docid: 105952
negative_passages.text: If each $a_n >0$ and $\sum a_n$ diverges, prove that $\sum a_n x^n \to +\infty$ as $x\to1^-$.
negative_passages.docid: 16143
negative_passages.text: How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
negative_passages.docid: 31889
negative_passages.text: The limit as $n$ approaches infinity of $n(a^{1/n}-1)$
negative_passages.docid: 140458
negative_passages.text: Infinite summation involving exponents
negative_passages.docid: 85465
negative_passages.text: Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
negative_passages.docid: 4253
negative_passages.text: How to calculate: $\sum_{n=1}^{\infty} n a^n$
negative_passages.docid: 115411
negative_passages.text: if $A_n \longrightarrow \infty $ and $B_n \longrightarrow \infty $ then $(A_n+B_n) \longrightarrow \infty$
negative_passages.docid: 91523
negative_passages.text: Prove this limit as $x\to\infty$
negative_passages.docid: 127627
negative_passages.text: Does $\sum^{\infty}_{n=1} n/(n-1)!$ converge to something?
negative_passages.docid: 66441
negative_passages.text: Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
negative_passages.docid: 113384
negative_passages.text: $\sum_{n\geq1} k_n x_n$ converges for every $x \in l_1$, then $\sup_{n\geq1} |k_n| < \infty$
negative_passages.docid: 61696
negative_passages.text: Prove that $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ converges when $n \to \infty$
negative_passages.docid: 74204
negative_passages.text: Evaluate $\lim_{n\to\infty}(1+x)(1+x^2)\cdots(1+x^{2n}),|x|<1$
negative_passages.docid: 25852
negative_passages.text: Does the series $\sum_{n=1}^\infty$ ${\sqrt{n+1}-\sqrt{n}}\over n$ converge or diverge?
negative_passages.docid: 133886
negative_passages.text: Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$