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Chapter 8: Problem 21
Show that if \(\left(f_{n}\right),\left(g_{n}\right)\) converge uniformly on the set \(A\) to \(f, g\), respectively, then \(\left(f_{n}+g_{n}\right)\) converges uniformly on \(A\) to \(f+g\).
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Let \(f_{n}(x):=x^{n} / n\) for \(x \in[0,1] .\) Show that the sequence \(\left(f_{n}\right)\) of differentiable functions converges uniformly to a differcntiable function \(f\) on \([0,1]\), and that the sequence \(\left(f_{n}^{\prime}\right)\) converges on \([0,1]\) to a function \(g\), but that \(g(1) \neq f^{\prime}(1)\).
Calculate \(\cos (.2), \sin (.2)\) and \(\cos 1, \sin 1\) correct to four decimal places.
If \(x \geq 0\) and \(n \in \mathbb{N}\), show that $$ \frac{1}{x+1}=1-x+x^{2}-x^{3}+\cdots+(-x)^{n-1}+\frac{(-x)^{n}}{1+x} $$ Use this to show that $$ \ln (x+1)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots+(-1)^{n-1} \frac{x^{n}}{n}+\int_{0}^{x} \frac{(-t)^{n}}{1+t} d t $$
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 \(\mathbb{R}\).
If \(a>0, a \neq 1\), and \(x\) and \(y\) belong to \((0, \infty)\), prove that \(\log _{a}(x y)=\log _{a} x+\log _{a} y\).
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