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Is this true?

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

No — it is false.

The counterexample

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

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

The kind of mistake this is

Boundary and endpoint

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

Drill statements like this

series power series

Others that fail the same way

From Sequences and Series of Functions › Power series, radius of convergence, Abel's theorem

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