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absolute convergence of integral and boundedness of derivative
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(Theorem)
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Theorem. Assume that we have an absolutely converging integral $$\int_a^\infty\!f(x)\,dx$$ where the real function $f$ and its derivative $f'$ are continuous and $f'$ additionally bounded on the interval $[a,\,\infty)$ . Then
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(1) |
Proof. If $c > a$ , we obtain $$\int_a^c\!f(x)f'(x)\,dx \;=\; \frac{1}{2}\sijoitus{a}{\quad c}\!(f(x))^2 \;=\; \frac{(f(c))^2-(f(a))^2}{2},$$ from which
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(2) |
Using the boundedness of $f'$ and the absolute convergence, we can estimate upwards the integral $$\int_a^c\!|f(x)f'(x)|\,dx \;=\; \int_a^c\!|f(x)||f'(x)|\,dx \;\leqq\; M\!\int_a^c\!|f(x)|\,dx \;\leqq\; M\!\int_a^\infty\!|f(x)|\,dx \quad \forall c \in [a,\,\infty)$$ whence $\int_a^\infty\!|f(x)f'(x)|\,dx$ is finite and thus $\int_a^\infty\!f(x)f'(x)\,dx$ converges absolutely. Hence (2) implies
$$\lim_{c\to\infty}(f(c))^2 \;=\; (f(a))^2+2\int_a^\infty\!f(x)f'(x)\,dx,$$ i.e. $\displaystyle\lim_{x\to\infty}(f(x))^2$ exists as finite, therefore also $$\lim_{x\to\infty}|f(x)| \;:=\; A.$$ Antithesis: $A > 0$ . It implies that there is an $x_0\;(\geqq a)$ such that $$|f(x)| \;\geqq\; \frac{A}{2} \quad \forall x \geqq x_0.$$ If now $b > x_0$ , then we had $$\int_{x_0}^b\!|f(x)|\,dx \;\geqq\; \frac{A}{2}(b\!-\!x_0) \;\; \longrightarrow \infty \quad \mbox{as}\;\; b \to \infty.$$ This means that $\int_{x_0}^\infty|f(x)|\,dx$ and consequently also $\int_a^\infty|f(x)|\,dx$ would be divergent. Since it is not true, we infer
that $A = 0$ , i.e. that the assertion (1) is true.
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"absolute convergence of integral and boundedness of derivative" is owned by pahio.
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Cross-references: divergent, implies, converges absolutely, finite, estimate, absolute convergence, proof, interval, bounded, continuous, derivative, real function, integral, theorem
This is version 4 of absolute convergence of integral and boundedness of derivative, born on 2009-09-03, modified 2009-09-03.
Object id is 11896, canonical name is AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative.
Accessed 278 times total.
Classification:
| AMS MSC: | 40A10 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of integrals) |
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Pending Errata and Addenda
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