91Ó°ÊÓ

Prove that if \(a, b\) are strictly positive integers then $$ \frac{a^{2}}{b^{2}}<2 \Rightarrow \frac{(a+2 b)^{2}}{(a+b)^{2}}<2 $$ Prove, moreover, that $$ \frac{(a+2 b)^{2}}{(a+b)^{2}}-2<2-\frac{a^{2}}{b^{2}} $$ This means that \(\frac{(a+2 b)^{2}}{(a+b)^{2}}\) is closer to 2 than \(\frac{a^{2}}{b^{2}}\) is.

Demonstrate that $$ \sum_{k=0}^{n}(k-n p)^{2}\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k}=n p(1-p) $$

Prove that \(A \subseteq B\) and \(\quad C \subseteq B \Longrightarrow A \cup C \subseteq B .\)

Prove that for all integers \(n \geq 0\) the inequality \(\boldsymbol{n}(n-\) 1) \(<2^{n+1}\) is verified.

Prove that \(f: x \rightarrow \frac{x-1}{x+1}\) is a bijection and find \(f^{-1} .\)

Let \(f^{[1]}(x)=f(x)=x+1, f^{[n+1]}=f \circ f^{[n]}, n \geq 1 .\) Find a closed formula for \(\boldsymbol{f}^{[n]}\) (1) \(^{n}\)

Let \(\boldsymbol{x} \in \mathbb{R} \backslash\\{\mathbf{1}\\}\) and let \(\boldsymbol{n} \in \mathbb{N} \backslash\\{\mathbf{0}\\}\). Prove that \(\sum_{k=0}^{n} \frac{2^{k}}{x^{2^{k}}+1}=\frac{1}{x-1}-\frac{2^{n+1}}{x^{2^{n+1}}+1}\)

Shew that \(\forall x>0, x\) is farther from \(\sqrt{5}\) than \(\frac{2 x+5}{x+2}\) is.

Prove that \(\forall(a, b, c) \in \mathbb{R}^{3}\), $$ a^{2}+b^{2}+c^{2} \geq a b+b c+c a $$

Problem 1.1.10 Prove the following distributive laws: \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C), \quad A \cup(B \cap C)=(A \cup B) \cap(A \cup C)\)