Is the following statement true? Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L

Counterexample

Is this true?

$\sum{}a_{n}x^{n} \to L$ as x → 1⁻ $\Rightarrow \sum{}a_{n} = L$

No — it is false.

The counterexample

$\sum(-1)^{n}x^{n} = 1/(1 + x) \to 1/2$

Why it fails

$\sum(-1)^{n}$ diverges. Abel's theorem has no converse without a Tauberian condition.

Keep practicing

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.