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Chapter 8: Problem 7
Show that \(\ln (e / 2)=1-\ln 2\). Use this result to calculate \(\ln 2\) accurate to four decimal places.
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Evaluate \(\lim \left(x^{n} /\left(1+x^{n}\right)\right)\) for \(x \in \mathbb{R}, x \geq 0\)
Prove that if \(a>0, a \neq 1\), then \(a^{\log _{a} x}=x\) for all \(x \in(0, \infty)\) and \(\log _{a}\left(a^{y}\right)=y\) for all \(y \in \mathbb{R}\). Therefore the function \(x \mapsto \log _{a} x\) on \((0, \infty)\) to \(\mathbb{R}\) is inverse to the function \(y \mapsto a^{y}\) on \(\mathbb{R}\).
Let \(a_{k}>0\) for \(k=1, \cdots, n\) and let \(A:=\left(a_{1}+\cdots+a_{n}\right) / n\) be the arithmetic mean of these numbers. For each \(k\), put \(x_{k}:=a_{k} / A-1\) in the inequality \(1+x \leq e^{x}(\) valid for \(x \geq 0)\). Multiply the resulting terms to prove the Arithmetic-Geometric Mean Inequality (6) $$ \left(a_{1} \cdots a_{n}\right)^{1 / n} \leq \frac{1}{n}\left(a_{1}+\cdots+a_{n}\right) $$ Moreover, show that equality holds in \((6)\) if and only if \(a_{1}=a_{2}=\cdots=a_{n}\).
Suppose the sequence \(\left(f_{n}\right)\) converges uniformly to \(f\) on the set \(A\), and suppose that each \(f_{n}\) is bounded on \(A\). (That is, for each \(n\) there is a constant \(M_{n}\) such that \(\left|f_{n}(x)\right| \leq M_{n}\) for all \(x \in A\).) Show that the function \(f\) is bounded on \(A\).
Show that \(|\sin x| \leq 1\) and \(|\cos x| \leq 1\) for all \(x \in \mathbb{R}\)
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