A New Method for Finding the Numerical Sum of an Infinite Series

1933 ◽  
Vol 40 (9) ◽  
pp. 537 ◽  
Author(s):  
C. C. Camp
Keyword(s):  
Infinite Series ◽  
New Method

XX. A new method of investigating the sums of infinite series

1791 ◽  
Vol 81 ◽  
pp. 295-316
Keyword(s):  
Infinite Series ◽  
New Method

The summation of infinite series is a subject, not only of curious speculation, but also of the greatest importance in the various branches of mathematics and philosophy; in consequence of which it has always claimed a very considerable share of attention from the most celebrated mathematicians.

Further Contributions to the Theory of Peripheral Jets in Ground Proximity

1992 ◽  
Vol 36 (01) ◽  
pp. 88-90
Author(s):  
David S. Tselnik
Keyword(s):  
General Solution ◽  
Infinite Series ◽  
Elliptic Functions ◽  
Jet Flow ◽  
New Method ◽  
Asymptotic Formulas ◽  
Flow Problems ◽  
Complicated Task

A number of plane inviscid jet flow problems of interest in hydrodynamics require the use of elliptic functions theory. Generally speaking, finding the general solution to a problem in terms of elliptic functions is not a complicated task. However, finding solutions as rapidly convergent infinite series or as sound asymptotic formulas is often not as easy, and special ways of treatment may prove to be necessary. In parallel with solving the problem of peripheral jets, the author's earlier paper (1985) proposed some such ways of treatment. In the present paper, a new method of treatment is proposed (and used);this approach may be of help in studies where the methods of elliptic functions theory have to be used.

Some applications of conformal transformation to airscrew theory

1936 ◽  
Vol 32 (4) ◽  
pp. 676-684 ◽  
Author(s):  
F. L. Westwater
Keyword(s):  
Infinite Series ◽  
Conformal Transformation ◽  
Two Dimensions ◽  
New Method

ABSTRACTRecent reports contain tables of a parameter K required in calculating the performance of an airscrew by a new method.The method of calculating K (due to Goldstein) is unsuitable for large values of the pitch especially near the tip of the airscrew.In the case of infinite pitch we fall back, for a two-bladed airscrew, on the problem of a rotating lamina in two dimensions.The solution for a cross lamina (corresponding to a four-bladed propeller) is given below and the tables of K for four blades are completed.A formula for the limit of K/Kp at the airscrew tip is given for a propeller with any number of blades, where Kp is an approximate value of K due to Prandtl.K for any number of blades is given in the form of an infinite series. The case of three blades is discussed in detail.

scholarly journals XXV. A new method of investigating the sums of infinite series. By the Rev. S. Vince, A.M. of Cambridge, in a letter to Henry Maty, A.M. Secretary

1782 ◽  
Vol 72 ◽  
pp. 389-416
Keyword(s):  
Infinite Series ◽  
Royal Society ◽  
New Method ◽  
General Will ◽  
The Third ◽  
The Subject ◽  
General Method

Sir, Having lately discovered some very easy methods of investigating the sums of certain infinite series, I have taken the liberty of requesting the favour of you to present them to the Royal Society. I have divided the subject into three parts: the first contains a new and general method of finding the sum of those series which De Moivre has found in one or two particular cases; but whose method, although it be in appearance general, will, upon trial, be found to be absolutely impracticable. The second contains the summation of certain series, the last differences of whose numerators become equal to nothing. The third contains observations on a correction which is necessary in investigating the sums of certain series by collecting two terms into one, with its application to a variety of cases.

Strong summability of infinite series on a scale of Abel type summability methods

1967 ◽  
Vol 63 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Babban Prasad Mishra
Keyword(s):  
Infinite Series ◽  
Open Interval ◽  
Strong Summability ◽  
New Method ◽  
Finite Limit ◽  
Summability Methods ◽  
Image Position

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Letand let {sn} be any sequence of numbers. If, for λ > − 1,is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.

XI. A New Method of expressing the Coefficients of the Development of the Algebraic Formula (a2 + b2 − 2ab cos φ)n, by means of the Perimeters of two Ellipses, when n denotes the Half of any Odd Number; together with an Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 253-293
Author(s):  
William Wallace
Keyword(s):  
Infinite Series ◽  
Small Degree ◽  
New Method ◽  
Moderate Degree ◽  
The Sun ◽  
Algebraic Formula ◽  
The Subject ◽  
Mutual Action

In calculating the effect of the mutual action of two planets upon each other, it has been found necessary to develop the algebraic formula (a2 + b2 — 2ab cos φ)n into a series of this form, A + B cos φ + C cos 2φ + D cos 3φ + &c. Here a and b denote the distances of the planets from the sun; φ denotes the angle of commutation; and the values of n, more immediately the subject of consideration, are —, and —.The determination of the coefficients A, B, C, &c. in these cases, appears to have been considered as a matter of difficulty by the mathematicians who first applied to the solution of the problem; for they found, that although it was only necessary to compute the first two coefficients A and B, the rest being easily derived from them, yet it did not appear that they could be expressed in finite terms, nor even by means of circular arches, or by logarithms. Recourse was therefore had to other methods, and chiefly to the method of infinite series; but as the series which most readily occurred to them, converged in some cases so slowly as to be in a manner useless, no small degree of analytical address has been found necessary, either to render it more convergent, or to find the sum of a competent number of its terms, with a moderate degree of labour.

Quantum field theory of bound states. I. Bound states in weak interaction

1952 ◽  
Vol 215 (1120) ◽  
pp. 133-146 ◽  
Keyword(s):  
Field Theory ◽  
Weak Interaction ◽  
Infinite Series ◽  
Bound States ◽  
Bound State ◽  
New Method ◽  
Scattering Method ◽  
Fundamental Particles ◽  
State Method ◽  
General Conditions

When compound particles as well as fundamental particles are involved in a collision two different mathematical techniques are required. These are (i) the scattering method as developed in the field theory of the S matrix, and (ii) the bound-state method which corresponds to a selective summation of infinite series of terms in the S matrix. This correspondence is discussed and general conditions are obtained which must be satisfied by any bound-state method. Weak interaction assumptions are made which permit the complete separation of the bound-state method and the scattering method in a collision involving bound states. The separation is achieved by a modified interaction representation in which a constant-state vector may describe fundamental particles and one or more compound particles moving without mutual interaction. The calculation of the S matrix between two states which are constant in this representation involves the propagation functions for compound particles. These propagators are discussed from two viewpoints, that of Bethe & Salpeter (1951), and a new method based on the modified interaction representation. These methods differ in the way in which infinite series of interactions are summed, though both satisfy the general conditions developed earlier in the paper. It is shown that, for the two-body problem in quantum electrodynamics, the new method leads to a propagator which contains a complex rest mass term. This is the relativistic analogue of the complex energy which Dirac (1947) has related to spontaneous emission. The method is illustrated by a simple example on resonance scattering.

THE APPLICATION OF FINITE FORWARD DIFFERENCES IN THE RESISTIVITY COMPUTATIONS OVER A LAYERED EARTH

Geophysics ◽  
1966 ◽  
Vol 31 (5) ◽  
pp. 971-980 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Infinite Series ◽  
New Method ◽  
Layered Earth ◽  
Truncation Errors ◽  
Resistivity Method ◽  
Computational Labor ◽  
Forward Differences ◽  
Correct Estimate

The paper describes a new method of computing the infinite series which arise in the calculation of standard graphs in the resistivity method. It is shown that, by using finite forward differences of higher orders and repeated summations by parts, the convergence of the potential and resistivity series can be improved appreciably. The procedure allows calculations with any degree of accuracy and with a substantial savings in computational labor. A correct estimate of the truncation errors involved may easily be determined.

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