This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...at a finite number of points in any interval, however large. For example, let there be infinite discontinuities only at xx, x2, ... xn in x = a, f(x) being bounded in any interval (c, b), where ca; . Let atkxx....? &. Then we have, as above ( 59), "f(x)dx= Xif(x)dx+ f(x)dx+...+ fx)dx+ "f(x)dx, where xnC.cib, provided ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...at a finite number of points in any interval, however large. For example, let there be infinite discontinuities only at xx, x2, ... xn in x = a, f(x) being bounded in any interval (c, b), where ca; . Let atkxx....? &. Then we have, as above ( 59), "f(x)dx= Xif(x)dx+ f(x)dx+...+ fx)dx+ "f(x)dx, where xnC.cib, provided that the integrals on the right-hand side exist. It will be noticed that the last integral I f(x) dx is an ordinary integral, f(x) being bounded and integrable in (c, b). If the integral f(x) dx also converges, we define the infinite poo J C integral I f(x)dx by Hie equation: J a f(x)dx=: if(x)dx+y(x)dx+...+ ' fx)dx+ f(x)dx. It is clear that this definition is independent of the position of c, since we have ' fx)dx+ fx)dx=C f(x)dx+ f(x)dx, J Xn J C J Xn J r' where Also we may write the above in the form f f(x)dx=' f(z)dx+2 f(x)dx+... + f(x)dx. J a J a J xx J xH The verbal alterations required in the definition of J f(x) dx 00 J-tO are obvious, and we define I f(x)dx, as before, as the sum of J-00 the integrals I f(x) dx and I f(x). J-x J a It is easy to show that this definition is independent of the position of the point a. 61. Tests for Convergence of f f(x) dx. It is clear that we J a need only discuss the case when there is a point of infinite discontinuity at an end of the interval of integration. If x = a is the only point of infinite discontinuity, we have "f(x) dx= Lt f fx)dx, Ja f-XJi+f when this limit exists. It follows at once, from the definition, that: I. The integral f f(x) dx is convergent and has the value I Ja when, any 'positive number e having been chosen, as small as we please, there is a positive number tj such that It follows from I. that, if I f(x) dx converges, to the arbitrary J a posi...
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