query: Find the least Positive Integer Satisfying $1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} \ge 4$
positive_passages.docid: 155951
positive_passages.text: Harmonic series sum approximation
negative_passages.docid: 68725
negative_passages.text: Find expression for : $ S_n =\sum_{i=1}^{n} \frac{i}{i^4+i^2+1} $
negative_passages.docid: 112005
negative_passages.text: Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 46661
negative_passages.text: A binomial inequality with factorial fractions: $\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$
negative_passages.docid: 14648
negative_passages.text: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer
negative_passages.docid: 27859
negative_passages.text: least common multiple of $\{1,2,...,n\}$ is bigger than $2^{n-1}$
negative_passages.docid: 5332
negative_passages.text: Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
negative_passages.docid: 8383
negative_passages.text: Are there any positive integers such that $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1$?
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: 137522
negative_passages.text: Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $
negative_passages.docid: 141357
negative_passages.text: Is $1+2+3+4+\cdots=-\frac{1}{12}$ the unique ''value'' of this series?
negative_passages.docid: 45171
negative_passages.text: How prove this $(p-1)!\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\right)\equiv 0\pmod{p^2}$
negative_passages.docid: 47072
negative_passages.text: Proving the inequality $\frac{9}4\ + \frac{3abc}4\ \ge ab+bc+ca$
negative_passages.docid: 108238
negative_passages.text: Find the smallest integer $n$ greater than $1$ such that the last $3$ digits of $n^2$ are the same as the last $3$ digits of $n$.
negative_passages.docid: 76652
negative_passages.text: Finding the maximum $lcm$ of a set of numbers with sum $n$
negative_passages.docid: 33314
negative_passages.text: How many non-negative integer solutions does $x_1+x_2+\cdots+x_n=A$ have?