By E. T. Whittaker, G. N. Watson
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Publish 12 months 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 offers an creation to the final conception of endless methods and of analytic capabilities including an account of the primary 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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Additional resources 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)
Let e be an arbitrary positive number, and let n be chosen so that I Un+1 \ + \Un 2 I + • • • + I Un I < - € for all values of p. Suppose that in order to obtain the first n terms of S we have to take m terms of 8f; then if k > m, Sk = Sn 4- terms of S with suffices greater than n, so that Sk —S = Sn — S + terms of S with suffices greater than n. Now the modulus of the sum of any number of terms of S with suffices greater than n does not exceed the sum of their moduli, and therefore is less than g €• Therefore But \Sk -S\ < | S n - S | + L .
Thus (writing £e for e) a necessary condition for uniformity of convergence is that | S (z) — Sn (z) | < e, whenever n>N and N is independent of z; the condition is also sufficient] for if it is satisfied it follows as in § 222 (I) that | RNiP(z) | < 2e, which, by definition, is the condition for uniformity. Example 1. Shew that, if x be real, the sum of the series \ '" • • • I t/ ' l \ i "I"} tf ii^ i • • • is discontinuous at x=0 and the series is non-uniformly convergent near #=0. The sum of the first n terms is easily seen to be 1 sum is 0; when #4=0, the sum is 1.
The convergence of the product in which an_i = - 1/n2 was investigated by Wallis as early as 1655. f See the Appendix, § A-2. 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.
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) by E. T. Whittaker, G. N. Watson