query: How to reach $\dfrac{(n-1)n(2n-1)}{6n^3}$
positive_passages.docid: 16812
positive_passages.text: Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
negative_passages.docid: 121128
negative_passages.text: Calculating a Residue at one of the poles of $\frac{1}{1+z^n}$
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: 17766
negative_passages.text: Find $n^{th}$ term of $\frac23,\frac35,\frac58,\frac8{13}\dots$
negative_passages.docid: 23726
negative_passages.text: A closed form for $1^{2}-2^{2}+3^{2}-4^{2}+ \cdots + (-1)^{n-1}n^{2}$
negative_passages.docid: 61858
negative_passages.text: Is there $\phi(n)=n/6$
negative_passages.docid: 133819
negative_passages.text: How to prove that $364 \mid n^{91} - n^7$?
negative_passages.docid: 153269
negative_passages.text: Find $\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$
negative_passages.docid: 52857
negative_passages.text: Behaviour of $\frac{x^{3} y}{x^{6} + y^{2}}$ around origin
negative_passages.docid: 68824
negative_passages.text: Show that $\frac{(3^{77}-1)}{2}$ is odd and composite
negative_passages.docid: 96039
negative_passages.text: Why is $\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}\le \frac 1 {n+1}$
negative_passages.docid: 4024
negative_passages.text: Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
negative_passages.docid: 94958
negative_passages.text: Prob. 6 (d), Chap. 1, in Baby Rudin, 3rd ed: How to complete this proof?
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$
negative_passages.docid: 76977
negative_passages.text: Prove that $f(n)$ is the nearest integer to $\frac12(1+\sqrt2)^{n+1}$?
negative_passages.docid: 68561
negative_passages.text: How to show $\lceil ( \sqrt3 +1)^{2n}\rceil$ where $n \in \mathbb{ N}$ is divisible by $2^{n+1}$
negative_passages.docid: 119146
negative_passages.text: How to solve $n < 2^{n/8}$ for $n$?