query: Continued fraction for $[1,2,3,4,5,6,\dots]$
positive_passages.docid: 19633
positive_passages.text: Closed form for a pair of continued fractions
negative_passages.docid: 128415
negative_passages.text: Prove that $1<\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\dots+\frac{1}{3001}<\frac43$
negative_passages.docid: 60385
negative_passages.text: How to solve this sequence $165,195,255,285,345,x$
negative_passages.docid: 111822
negative_passages.text: Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?
negative_passages.docid: 110262
negative_passages.text: Finding a closed formula for $1\cdot2\cdot3\cdots k +\dots + n(n+1)(n+2)\cdots(k+n-1)$
negative_passages.docid: 22505
negative_passages.text: Evaluate : $1+\frac{2}{6}+\frac{2\cdot 5}{6\cdot 12}+\frac{2\cdot 5\cdot 8}{6\cdot12\cdot 18}+\ldots$
negative_passages.docid: 18814
negative_passages.text: Sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$
negative_passages.docid: 73570
negative_passages.text: Sum of (example: 0+1+2+3 = 6 , 0+1+2+3+4+5+6+7 = 28 and so on)
negative_passages.docid: 83910
negative_passages.text: Integer part of $\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$
negative_passages.docid: 102425
negative_passages.text: Limit $ a_{n+1} = \frac{1}{3}(2a_n + \frac{5}{a_n^{2}}), a_1 > 0 $
negative_passages.docid: 136459
negative_passages.text: Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
negative_passages.docid: 115162
negative_passages.text: Establish convergence of the series: $1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-...$
negative_passages.docid: 61871
negative_passages.text: Formula for the $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5, ...$
negative_passages.docid: 109059
negative_passages.text: Evaluate $\lim\limits_{n\to \infty}\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{6n}$
negative_passages.docid: 90074
negative_passages.text: Show that the sequence $1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,\cdots$ isn't periodic
negative_passages.docid: 153796
negative_passages.text: How to define fractional factorials, like 3.6!?
negative_passages.docid: 95786
negative_passages.text: Sum $\frac{1}{6} + \frac{5}{6\cdot 12} + \frac{5\cdot 8}{6\cdot 12\cdot 18} + \frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\ldots$