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Chapter 10: Problem 11
Show that if \(n \in \mathbb{N}, s>0\), then \(\int_{0}^{\infty} x^{n} e^{-s x} d x=n ! / s^{n+1}\).
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Ler \(f\) and \(|f|\) belong to \(R^{*}[a, \gamma]\) for every \(\gamma \geq a\). Show that \(f \in \mathcal{L}[a, \infty)\) if and only if the set \(V:=\left\\{\int_{a}^{x}|f|: x \geq a\right\\}\) is bounded in \(\mathbb{R}\).
Determine whether the following integrals are convergent or divergent. (Define the integrands to be 0 where they are not already defined.) (a) \(\int_{0}^{1} \frac{\sin x d x}{x^{3 / 2}}\), (b) \(\int_{0}^{1} \frac{\cos x d x}{x^{3 / 2}}\). (c) \(\int_{0}^{1} \frac{\ln x d x}{x \sqrt{1-x^{2}}}\), (d) \(\int_{0}^{1} \frac{\ln x d x}{1-x}\), (c) \(\int_{0}^{1}(\ln x)(\sin (1 / x)) d x\), (f) \(\int_{0}^{1} \frac{d x}{\sqrt{x}(1-x)}\).
Let \(g_{n}(x):=-1\) for \(x \in[-1,-1 / n)\). let \(g_{n}(x):=n x\) for \(x \in[-1 / n, 1 / n]\) and let \(g_{n}(x):=1\) for \(x \in(1 / n, 1]\). Show that \(\left\|g_{m}-g_{n}\right\| \rightarrow 0\) as \(m, n \rightarrow \infty\), so that the Completeness Theorem \(10.2 .12\) implies that there exists \(g \in \mathcal{L}[-1,1]\) such that \(\left(g_{n}\right)\) converges to \(g\) in \(\mathcal{L}[-1,1]\). Find such a function \(g\).
Consider the following sequences of functions with the indicated domains. Does the sequence converge? If so, to what? Is the convergence uniform? Is it bounded? If not bounded, is it dominated? Is it monotone? Evaluate the limit of the sequence of integrals. (a) \(\frac{k x}{1+k x}\) \([0,1]\) (b) \(\frac{x^{k}}{1+x^{k}} \quad[0,2]\), (c) \(\frac{1}{1+x^{k}}\) \([0,1]\), (d) \(\frac{1}{1+x^{k}}\) \([0,2]\)
Let \(\Phi(x):=x|\cos (\pi / x)|\) for \(x \in(0,1)\) and let \(\Phi(0):=0 .\) Then \(\Phi\) is continuous on \([0,1]\) and \(\Phi^{\prime}(x)\) exists for \(x \notin E:=\\{0\\} \cup\left\\{a_{k}: k \in \mathbb{N}\right\\}\), where \(a_{k}:=2 /(2 k+1) .\) Let \(\varphi(x):=\Phi^{\prime}(x)\) for \(x \notin E\) and \(\varphi(x):=0\) for \(x \in E\). Show that \(\varphi\) is not bounded on \([0,1]\). Using the Fundamental Theorem \(10.1 .9\) with \(E\) countable, conclude that \(\varphi \in \mathcal{R}^{*}[0,1]\) and that \(\int_{a}^{b} \varphi=\Phi(b)-\Phi(a)\) for \(a, b \in[0,1] .\) As in Exercise 19, show that \(|\varphi| \notin \mathcal{R}^{*}[0,1]\).
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