By E. A. Maxwell

ISBN-10: 0521056977

ISBN-13: 9780521056977

This is often the second one of a sequence of 4 volumes masking all levels of improvement of the Calculus, from the final years in school to measure general. The books are written for college students of technological know-how and engineering in addition to for expert mathematicians, and are designed to bridge the space among the works utilized in colleges and extra complicated stories. with their emphasis on rigour. This remedy of algebraic and trigonometric capabilities is the following built to hide logarithmic, exponential and hyperbolic features and the growth of these kind of features as energy sequence. there's a bankruptcy on curves and the assumption of complicated numbers is brought for the 1st time. within the ultimate chapters, the writer starts off a scientific remedy of equipment of integrating services, introducing rules into what usually turns out fairly a haphazard technique. This quantity, just like the others, is definitely endowed with examples.

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Y-y so that u n y- y, n +1 |^w+i| < \ n \y ~~ ° Since uno is a definite ascertainable number, and since \y\< I, it follows that as 7&->oo. But \Rn\oo, and the validity of the expansion is established. 10. The logarithmic series. -l&+ 71 This expansion for log (1 +x) is valid for all values of x in the range When x = 1, we have the result The case x = — 1 is reflected in the graph (Fig. 62, p. 4) where y-> —oo as #->0. EXAMPLE IV I.

61 5! 9 by not more than ^—j-; 39 /p5 z^7 #-^7+-£7-^ oI oI 71 and so on. 2tl1 whose nth term is ( — I)71-1 -— —-, with the property that sin# v^ - 1 ) 1 differs from the sum of the first n terms by less than Let us examine this 'difference' term j - ~ - — — , writing it in the form I I I I I I I I II (2n-{-1)! II ~F*~2~*~3~#~4~*"#"272 Suppose that x has some definite value, positive or negative. If we 'watch' n increase a step at a time, there will come a point when 2n+l exceeds \x\. Thereafter, the later factors in the product are less than 1; moreover the factor tends to zero as n continues to increase.

If we write 6 = a + h, Taylor's theorem (with the Lagrange remainder) becomes where f is a certain number between a,a + h. A convenient form is found by putting a = 0 and then renaming h to be the current variable x: f(x) = |J ^^ where f is a certain number between 0,x. This important result is known as Maclaurin's theorem. Compare p. 43. With the Cauchy form of remainder, the corresponding result is where ^ is a certain number between 0,1. 7* Maclaurin's series. The remainder Rn in Maclaurin's theorem appears in the form or 50 T A Y L O R ' S S E M E S AND A L L I E D R E S U L T S where £ lies between 09x and 6 between 0,1.

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An Analytical Calculus: Volume 2: For School and University (v. 2) by E. A. Maxwell

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