By E. T. Whittaker, G. N. Watson
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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 offers an creation to the overall conception of countless strategies and of analytic services 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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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)
II PRINGSHEIM'S THEOREM* : If 8 exists and the sums by rows and columns exist, then each of these sums is equal to S. For since S exists, then we can find m such that i, v > m. And therefore, since lim 8^ „ exists, | ( lim $Mt „) — SI ^ e; that is to say, when fi>m, and so (§ 2 22) the sum by rows converges to S. In like manner the sum by columns converges to S. 2*52. Absolutely convergent double series. We can prove the analogue of § 2*41 for double series, namely that if the terms of an absolutely convergent double series are taken in any order as a simple series, their sum tends to the same limit, provided that every term occurs in the summation.
T"'" converges absolutely if | z \<\ b | - 1 . [For - ^ = ^ z-*~-bz, as w-*-oo ; so the condition for absolute convergence is * Opuscules, t. v. (1768), pp. 171-182. 2-36, 2-37] Example 3. 23 THE THEORY OF CONVERGENCE Shew t h a t the series 2 , , _ ^ - converges absolutely if | s | < l . ^ - ( l + w 1 ) * ! > ( 1 +n~l)n-\z» [For, when \z\<\, n—1 \ > 1 + 1 + ^ p + --- ~ 1>1 » so the moduli of the terms of t h e series are less t h a n the corresponding terms of the series 2 n I zn~l I; b u t this latter series is absolutely convergent, and so the given series con/* = !
M_i>v - £,*_,,„_,), it is easily seen that, if the double series is convergent, then lim u^ „ = 0. Stolz necessary and sufficient^ condition for convergence. A condition for convergence which is obviously necessary (see § 2*22) is that, given e, we can find m and n such that | Stl+Pt v+a — $M> „ j < e whenever p > m and v > n and p, a may take any of the values 0, 1, 2, .... The condition is also sufficient; for, suppose it satisfied; then, when /jL>m + n, | ^ +P>M+P - $M|M | < e. Therefore, by §2*22, $M)M has a limit S; and then making p and a tend to infinity in such a way that /JL + p = v + a, we see that | S — $M> „ | ^ e whenever fi > m and v > n; that is to say, the double series converges.
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