By E. T. Whittaker, G. N. Watson
Filenote: PDF retail from ebsco. Please notice this is often PDF retail however the writer has scanned and serarchable textual content or even hyperlinked the index quantity. I checked ebl, ebrary and ebsco, all have a similar quality.
Publish yr note: First released in 1927
This vintage textual content is understood to and utilized by hundreds of thousands of mathematicians and scholars of arithmetic thorughout the realm. It supplies an creation to the overall idea of endless approaches and of analytic features including an account of the main transcendental functions.
Read Online or Download A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library) PDF
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Extra info for A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (4th Edition) (Cambridge Mathematical Library)
This follows from the fact that, the series T being convergent, it is always possible to choose n so that the remainder after the first n terms of T, and therefore the modulus of the remainder after the first n terms of S, is less than an assigned positive number e; and since the value of n thus found is independent of zy it follows (§ 3*31) that the series S is uniformly convergent; by § 2*34, the series S also converges absolutely. Example. The series CO8 Z + TTZ COS 2 Z+ —9 COS 3 Z -f . . is uniformly convergent for all real values of z, because the moduli of its terms are not greater than the corresponding terms of the convergent series 1+ 2~2 + 32+'"' whose terms are positive constants.
24 [CHAP. II THE PROCESSES OF ANALYSIS (—\ where Ax is independent of n, Corollary. If then the series is absolutely convergent if Ax < — 1. 00 n / 1\ Example. Investigate the convergence of 2 nr exp ( — k2 —),when r>k and when n=i r
27, 271] THE THEORY OF CONVERGENCE 33 Now, since lim an = 0, we can find m such that, when n > m, \ an \ < \ ; and then i an~l log (1 + a,,) - 1 | = i 1 -2 22 "•" 2 s ~*~ * " ~~ 2 ' log (1 + On) And thence, when n > m, - 3 ^; therefore, by the comparison theorem, the absolute convergence of 2 log (1 -f G^) entails that of %an and conversely, provided that an 4= — 1 for any value of n. This establishes the result*. If, in a product, a finite number of factors vanish, and if, when these are suppressed, the resulting product converges, the original product is said to converge to zero.
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