Transcription of Improper Integrals - Trinity University
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Improper Integrals - Trinity University
Improper IntegralsR. C. DailedaBy animproper integralwe will mean an integral of the form af(x)dx.(1)The goal of this note is to carefully define, and then study the properties of, improperintegrals. To this end, leta Rand letfbe a function that is Riemann integrable on everyfinite subinterval of [a, ). We then define af(x)dx= limA Aaf(x)dxprovided the limit on the right-hand side exists. In this case we say that the improperintegral (1)convergesand that its value is that of the limit. Otherwise we say (1) turns out that the theory of Improper Integrals closely mirrors that of infinite series, andwe will prove several results which can be viewed as direct analogues of results about example, one can use the usual limit laws to immediately verify that Improper Integrals (when convergent) obey the usual linearity of defi]
That’s the easy implication. For the converse, now suppose the stated Cauchy criterion holds. For natural numbers n alet a n = Z n a f(x)dx: Let …
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