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. 1916 Excerpt: ... un hm = x0 n--oo ujt--i and hence---= ito ] -n; hm tn = 0. "71--1 n=oo Whence un = u0(xo + tl)(xo + t2) (xQ + tn). Now, having chosen an arbitrarily small positive quantity 77, we have e r for all w a determinate value and hence o+ e = x0-r?; nnr Thus, as n increases indefinitely the expression u becomes infinite to ...
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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. 1916 Excerpt: ... un hm = x0 n--oo ujt--i and hence---= ito ] -n; hm tn = 0. "71--1 n=oo Whence un = u0(xo + tl)(xo + t2) (xQ + tn). Now, having chosen an arbitrarily small positive quantity 77, we have e r for all w a determinate value and hence o+ e = x0-r?; nnr Thus, as n increases indefinitely the expression u becomes infinite to as high an order as that of (: co-y)n-But for a sufficiently small choice of 77 we have a?o-1, since by hypothesis xo 1. Thus (55) cannot be satisfied for any value of r. In contrast to this result, we have the following important theorem arising when, instead of the definition (I) of sum, we adopt the definition (IV) of Borel. 4. Let 00 n=0 be any power series having a radius of convergence equal to 1. //, then, the series 00 a=0 is summable by definition (IV) ( 36) so also is the series 00 provided x0 lie within the polygon formed by tangents to the given circle at the points (assumed finite in number) upon the circumference at which f(x) has singularities. Moreover, f(x) may be extended analytically to all such points x0 by means of the sum formula in question, i. e., The summability at Xo will be absolute ( 42) and it will be uniform ( 43) throughout any region situated wholly within the indicated polygon (polygon of summability).32 5. Absolutely convergent series are absolutely summable, but series that are merely convergent may not be absolutely summable.33 6. // but one of two series is absolutely summable while both are summable by definition (IV) (Borel's integral) to the respective limits si, s2, then the product series (cf. (37)) is summable by the same definition to the value si, si, but not necessarily absolutely summable.34 7. // two series are summable by definition (IV) (Borel's integral)...
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Add this copy of Studies on Divergent Series and Summability, Volume 1 to cart. $48.02, good condition, Sold by Bonita rated 4.0 out of 5 stars, ships from Newport Coast, CA, UNITED STATES, published 2010 by Nabu Press.
