Partial Sums of Infinite Series, and How They Grow

1977 ◽  
Vol 84 (4) ◽  
pp. 237 ◽  
Author(s):  
R. P. Boas
Keyword(s):  
Infinite Series ◽  
Partial Sums

On the Absolute Nörlund Summability of a Fourier Series

1973 ◽  
Vol 16 (4) ◽  
pp. 599-602
Author(s):  
D. S. Goel ◽  
B. N. Sahney
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)

Possible and Impossible Series

2015 ◽  
Vol 108 (7) ◽  
pp. 560
Author(s):  
Mark MacLean
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Its Sequence

A lesson helps students discern possible relationships between an infinite series, its sequence of terms, and the sequence of partial sums.

On Absolute Summability by Riesz and Generalized Cesàro Means. I

1970 ◽  
Vol 22 (2) ◽  
pp. 202-208 ◽  
Author(s):  
H.-H. Körle
Keyword(s):  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Cesàro Means ◽  
Riesz Means ◽  
Cesaro Means ◽  
Cesáro Means ◽  
Image Position ◽  
Generalized Cesàro Means

1. The Cesàro methods for ordinary [9, p. 17; 6, p. 96] and for absolute [9, p. 25] summation of infinite series can be generalized by the Riesz methods [7, p. 21; 12; 9, p. 52; 6, p. 86; 5, p. 2] and by “the generalized Cesàro methods“ introduced by Burkill [4] and Borwein and Russell [3]. (Also cf. [2]; for another generalization, see [8].) These generalizations raise the question as to their equivalence.We shall consider series(1)with complex terms an. Throughout, we will assume that(2)and we call (1) Riesz summable to a sum s relative to the type λ = (λn) and to the order κ, or summable (R, λ, κ) to s briefly, if the Riesz means(of the partial sums of (1)) tend to s as x → ∞.

On a Tauberian theorem for absolute harmonic summability

1969 ◽  
Vol 65 (1) ◽  
pp. 93-99 ◽  
Author(s):  
O. P. Varshney ◽  
Govind Prasad
Keyword(s):  
Infinite Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Image Position ◽  
Harmonic Summability

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined bywhere is given by

Testing the hypothesis that a point is Poisson

1977 ◽  
Vol 9 (4) ◽  
pp. 724-746 ◽  
Author(s):  
Robert B. Davies
Keyword(s):  
Point Process ◽  
Infinite Series ◽  
Partial Sums ◽  
Test Statistic ◽  
Optimal Test ◽  
One Dimensional

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

scholarly journals Absolute summability of a fourier series and its derived series by a product method

1972 ◽  
Vol 14 (4) ◽  
pp. 470-481 ◽  
Author(s):  
H. P. Dikshit
Keyword(s):  
Fourier Series ◽  
Infinite Series ◽  
Absolute Summability ◽  
Partial Sums ◽  
Product Method ◽  
Derived Series

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let {Pn} be a sequence of constants, real or complex, and let us write Pn = p0 + p1 + … + pn; P-1 = P-1 = 0.

On the absolute Nörlund summability of a Fourier series

1967 ◽  
Vol 7 (2) ◽  
pp. 252-256 ◽  
Author(s):  
Fu Cheng Hsiang
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Partial Sums ◽  
The Absolute ◽  
Image Position ◽  
Absolute Nörlund Summability

Let Σn−0∞an, be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us writeIfas n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

A Tauberian theorem for absolute summability

1970 ◽  
Vol 67 (2) ◽  
pp. 321-326 ◽  
Author(s):  
G. Das
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Tauberian Theorem ◽  
Power Series Expansion ◽  
Binomial Coefficient ◽  
Absolute Summability ◽  
Partial Sums ◽  
Function Of Bounded Variation ◽  
Image Position ◽  
The Given

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by

Absolute total-effective (N, pn) means

1971 ◽  
Vol 69 (1) ◽  
pp. 107-122 ◽  
Author(s):  
H. P. Dikshit
Keyword(s):  
Infinite Series ◽  
Partial Sums ◽  
Image Position

1. Definitions and notations. Let be a given infinite series with the sequence of its partial sums {sn}. Let {pn} be a sequence of constants, real or complex, and let us write

scholarly journals Infinite series and the derived set of the aggregate of the fractional parts of its partial sums: Addendum

1982 ◽  
Vol 25 (2) ◽  
pp. 315-316
Author(s):  
S. Audinarayana Moorthy
Keyword(s):  
Infinite Series ◽  
General Term ◽  
Unit Interval ◽  
Partial Sums ◽  
Derived Set

In the author's paper [7] it was proved that the fractional parts of the partial sums of an infinite series (of real terms) diverging to +∞ or −∞, in which the general term tends to zero, are everywhere dense in the closed unit interval. This result was extended to series of infinite oscillation (see Remark 4.1 of the said paper) on the argument that a sequence of partial sums having infinite oscillation has a subsequence that diverges to +∞ or −∞.

Sign in / Sign up

Export Citation Format

Share Document