Applied Summability Methods by M. Mursaleen

Author:M. Mursaleen
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

7.2 Geometric Series

The following theorem is helpful in determining the region in which the sequence of partial sums of the geometric series is almost A-summable to .

Theorem 7.2.1.

Let D be a set of complex numbers with 0 ∈ D 0 and 1∉D. Let denote the sequence of partial sums of the geometric series. Then, s is almost A-summable to uniformly on each compact subset of D if and only if

(7.2.1)

(7.2.2)

uniformly on each compact subset of D.

Proof.

Suppose that (7.2.1) and (7.2.2) hold. Then,

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