query: Show that $\Sigma_{j=1}^{n} 1/j(j+1) = \dfrac{n}{n+1}$
positive_passages.docid: 11407
positive_passages.text: Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$
negative_passages.docid: 49806
negative_passages.text: Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$
negative_passages.docid: 111353
negative_passages.text: Show that $\{a_n\}$ defined by $a_{n+1}=\frac{a_n+2}{a_n+1}$ converges
negative_passages.docid: 8913
negative_passages.text: Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$
negative_passages.docid: 8192
negative_passages.text: Simple proof of showing the Harmonic number $H_n = \Theta (\log n)$
negative_passages.docid: 153226
negative_passages.text: Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
negative_passages.docid: 61670
negative_passages.text: Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$
negative_passages.docid: 73892
negative_passages.text: Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$
negative_passages.docid: 140614
negative_passages.text: Show that $\sum^\infty_{n=m}\frac{1}{n^2} = O(\frac{1}{m})$
negative_passages.docid: 53426
negative_passages.text: Proof of a formula involving Euler's totient function: $\varphi (mn) = \varphi (m) \varphi (n) \cdot \frac{d}{\varphi (d)}$
negative_passages.docid: 1712
negative_passages.text: $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
negative_passages.docid: 157898
negative_passages.text: Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$
negative_passages.docid: 83313
negative_passages.text: Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
negative_passages.docid: 112329
negative_passages.text: Show that $d(z,a)=|\frac{z-a}{1-\overline{a}z}|$ is a metric on the unit disk
negative_passages.docid: 89186
negative_passages.text: Showing that $\frac{X_{(i)}}{X_{(n)}},i=1,2,...,n-1$ and $X_{(n)}$ are independent for a population with df $F(y)=y^{\theta}$
negative_passages.docid: 137446
negative_passages.text: Show that if $b_k\uparrow \infty$ and $\Sigma_{k = 1}^\infty a_kb_k$ converges, then $b_m \Sigma_{k = m}^\infty a_k → 0$ as $m → \infty$.
negative_passages.docid: 91000
negative_passages.text: Prove that $\frac{1}{a+1}0$