query: How to prove that $n- \sqrt{n+a} \sqrt{n+b}$ converges to $-\frac{a+b}{2}$?
positive_passages.docid: 18550
positive_passages.text: Limit with multiple variables
negative_passages.docid: 59260
negative_passages.text: How to prove the convergence of $\sum_{n\geq 0}\frac{n!^d}{(dn)!}$ for $d\geq 2$?
negative_passages.docid: 4345
negative_passages.text: Prove that lim $(\sqrt{n^2+n}-n) = \frac{1}{2}$
negative_passages.docid: 12031
negative_passages.text: Asymptotic behavior of the sequence $a_n=\sqrt{n+a_{n-1}}$
negative_passages.docid: 32964
negative_passages.text: Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$
negative_passages.docid: 26500
negative_passages.text: Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge
negative_passages.docid: 40797
negative_passages.text: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
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: 84946
negative_passages.text: If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
negative_passages.docid: 59475
negative_passages.text: Show that $\sqrt{n+1}-\sqrt{n}\to0$
negative_passages.docid: 87834
negative_passages.text: How can we show that the following sequence converges?
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: 104344
negative_passages.text: Prove if $|a_{n+1}|/|a_n| \leq |b_{n+1}|/|{b_n}|$ and $\sum |b_n|$ is convergent, then $\sum |a_n|$ is convergent
negative_passages.docid: 18675
negative_passages.text: $\{a_n\}$ sequence $a_1=\sqrt{6}$ for $n \geq 1$ and $a_{n+1}=\sqrt{6+a_n}$ show it that convergence and also find $\lim_{x \to \infty} \{a_n\}$
negative_passages.docid: 122942
negative_passages.text: Prove that if $\sum |a_{n}|$ converges then $\sum a_{n}^{2}$ converges
negative_passages.docid: 20136
negative_passages.text: Convergence of the sequence $ \sqrt {n-2\sqrt n} - \sqrt n $
negative_passages.docid: 140220
negative_passages.text: Convergence of $\sum_{n=2}^{\infty}\frac{\sqrt{a_{n}}}{\ln\, n}(n^{a_{n}}-1)$