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A boundary limit theorem relating a convergent series to its associated power series near the radius 1.
Abel's theorem: Let be a convergent series of real or complex numbers, with sum . For , define
Then is well-defined for every , and
This connects ordinary convergence of series with boundary behavior of power series and complements radius-of-convergence results such as the Cauchy–Hadamard theorem.
A convergent series is a series whose partial sums converge to a finite limit , in which case one writes .
A power series is a series of the form with real coefficients and a real center .
There exists a number , called the radius of convergence, such that the series converges for and diverges for (the endpoint behavior at is decided separately). The radius can often be computed using the Cauchy–Hadamard theorem.
Cauchy–Hadamard theorem: For a power series (real or complex), define
Set the radius of convergence
with the conventions and . Then: