An Introduction to the Theory of Infinite SeriesMacmillan and Company, limited, 1908 - 511 pagine An Introduction to the Theory of Infinite Series by Thomas John I'Anson Bromwich, first published in 1908, is a rare manuscript, the original residing in one of the great libraries of the world. This book is a reproduction of that original, which has been scanned and cleaned by state-of-the-art publishing tools for better readability and enhanced appreciation. Restoration Editors' mission is to bring long out of print manuscripts back to life. Some smudges, annotations or unclear text may still exist, due to permanent damage to the original work. We believe the literary significance of the text justifies offering this reproduction, allowing a new generation to appreciate it. |
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An Introduction to the Theory of Infinite Series Thomas John I'Anson Bromwich,Thomas Murray MacRobert Visualizzazione estratti - 1942 |
An Introduction to the Theory of Infinite Series Thomas John I'Anson Bromwich Visualizzazione estratti - 1931 |
An Introduction to the Theory of Infinite Series Thomas John I'Anson Bromwich,Thomas Murray MacRobert Visualizzazione estratti - 1942 |
Parole e frasi comuni
a₁ Abel's Abel's Lemma Abel's test Abel's theorem absolutely convergent absolutely summable An+1 apply asymptotic series b₁ b₂ Borel's integral circle coefficients complex condition consequently continuous function convergent series converges absolutely converges uniformly decimal deduce definition denote differentiation Dirichlet's test divergent divergent series double series equal equation Euler's finite number fixed number follows formula given gives greater HARDY Hence increases inequality infinite interval last example less logarithmic Math method monotonic negative number of terms obtain oscillates positive integer positive number positive terms power-series Pringsheim proof prove radius of convergence rational numbers result satisfied sequence series converges shew Similarly sinh Suppose tends to infinity tends to zero theorem uniform convergence upper limit v₁ variable vergent write
Brani popolari
Pagina 253 - If two limiting processes performed in a definite order on a function of two variables lead to a definite value X, but when performed in reverse order lead to a meaningless expression Y, we may agree to interpret Y as meaning X.
Pagina 56 - X 106 electron volts, so that the ratio of the number of positive to the number of negative...
Pagina 252 - H will be 1/(1 + x), because the series arises from the expansion of the fraction, whatever number is put in place of x. If this is agreed, the new definition of the word sum coincides with the ordinary meaning when a series converges; and since divergent series have no sum in the proper sense of the word, no inconvenience can arise from this terminology. Finally, by means of this definition, we can preserve the utility of divergent series and defend their use from all objections.
Pagina 39 - Let an be a positive and steadily decreasing function of n, whose limit, as n->cc , is zero / and let pn be the number of positive terms and qn the number of negative terms in the first n terms of the series (1) ±al±a,±a3±..., so that pn+qn=n. Then if the series (1) is convergent, but not absolutely convergent, the ratio Pnjl...
Pagina 11 - The convergence or divergence of a series is not affected by the addition or omission of a finite number of terms at the beginning of the series, although the actual value of the limit will be changed.
Pagina 487 - From q39, q51 and the equation 02 + 02 + 12 + 12 = 2, prove that every positive integer can be expressed as the sum of four squares.
Pagina 252 - ... is obtained as the development of some closed expression, it may be used in mathematical operations as the equivalent of that expression, even for values of the variable for which the series diverges.
Pagina 202 - Now Tn is positive and the series which represents it is absolutely convergent and therefore its square forms a convergent series with the same radius of convergency. Now it is known (see Bromwich's Theory of Infinite Series, p. 216) that the circle of convergence of the reciprocal of a power-series is either the same as that of the original series, or else reaches up to the zero of the given series which is nearest to the origin. Thus...
Pagina 50 - It is to be observed that in the derangement we make a one-to-one correspondence between the terms of two series; so that every term in the first series occupies a perfectly definite place in the second series, and conversely. Thus, corresponding to any number (?(.) of terms...
Pagina 470 - V W' x sin (l/x) 20. The function which is equal to 1 when x is rational and to 0 when x is irrational (Ch. II, Ex. xvi. 10) is discontinuous for all values of x. So too is any function which is defined only for rational or for irrational values of x. 21. The function which is equal to x when x is irrational and to V{(1 +P*)/(1 + ?*)} when a; is a rational fraction p|q (Ch.