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Abel's limit theorem: a convergent complex series is recovered by its power series along every Stolz approach to 1

Statement

If the complex series $\sum_{n \geq 0} a_{n}$ converges to $s$ , then $F (z) = \sum_{n \geq 0} a_{n} z^{n}$ converges for $∣ z ∣ < 1$ and $F (z) ⟶ s (z \to 1)$ whenever $z$ remains in one fixed Stolz region Stolz approach regions at the boundary point 1 of the unit disc. In particular the conclusion holds for radial approach $z = r ↑ 1$ .

Facts & Assumptions

Given: A convergent complex series $\sum a_{n} = s$ , its partial sums, and a fixed $C \geq 1$ .

[L1]

Abel summation expresses a finite weighted sum in terms of partial sums and successive differences of the weights (Abel summation by parts for complex coefficients and their partial sums).

[L2]

Cauchy–Hadamard gives convergence inside the radius and makes no boundary assertion (Cauchy-Hadamard for complex power series, including zero and infinite radius).

[L3]

Complex modulus is multiplicative and satisfies the triangle inequality (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1

Replace $a_{0}$ by $a_{0} - s$ and write $t_{n} = \sum_{k = 0}^{n} a_{k}$ for the adjusted partial sums; then $t_{n} \to 0$ , and it suffices to prove that the adjusted power series tends to $0$ .

givenalgebra

1.2

For $∣ z ∣ < 1$ , [L1] followed by passage to the limit gives $\sum_{n \geq 0} a_{n} z^{n} = (1 - z) \sum_{n \geq 0} t_{n} z^{n}$ : the endpoint term $t_{N} z^{N}$ tends to $0$ , and convergence follows from boundedness of $(t_{n})$ and the geometric majorant, consistently with [L2].

L1L2L3

2.1

Given $ε > 0$ , choose $N$ with $∣ t_{n} ∣ < ε / (2 C)$ for $n \geq N$ . In step 1.2 split the sum before $N$ : the finite head times $1 - z$ tends to $0$ , while the tail has modulus at most $∣ 1 - z ∣ ε (2 C)^{- 1} \sum_{n \geq N} ∣ z ∣^{n} \leq ε / 2$ because $∣ 1 - z ∣ / (1 - ∣ z ∣) \leq C$ in the Stolz region.

step 1.2L3choose

3.1

Thus the adjusted series tends to $0$ , so the original tends to $s$ . The point $z = 1$ is used only as a limit endpoint, and radial approach is the case $C = 1$ .

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 77 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2, Theorem 3 (standard reference, not scraped)