query: How to prove the following equalities
positive_passages.docid: 154544
positive_passages.text: Prove that $ \sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}=\frac{z}{1-z} $
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
negative_passages.docid: 29462
negative_passages.text: Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
negative_passages.docid: 128189
negative_passages.text: Proving Integral Inequality
negative_passages.docid: 14386
negative_passages.text: Proof of an equality involving cosine $\sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{2}}}}\ =\ 2\cos (\pi/2^{n+1})$
negative_passages.docid: 5927
negative_passages.text: How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$
negative_passages.docid: 73987
negative_passages.text: Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
negative_passages.docid: 27954
negative_passages.text: Prove triangle inequality using the properties of absolute value
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 1932
negative_passages.text: How to show this equality of $\rho(X,Y)$
negative_passages.docid: 76347
negative_passages.text: Shorter proof for some equvalences
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 134049
negative_passages.text: Showing an inequality is valid
negative_passages.docid: 22237
negative_passages.text: Proof of triangle inequality
negative_passages.docid: 113943
negative_passages.text: Prove this equality by using Newton's Binomial Theorem
negative_passages.docid: 81775
negative_passages.text: Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$
negative_passages.docid: 49296
negative_passages.text: Prove $|(1+|u|^2)v-(1+|v|^2)u|>|u\bar{v}-\bar{u}v|$