The bookUnit 1 · Sequences and Series of Functions10 / 83

Power series, radius of convergence, Abel's theorem

Why this is asked: Radius via limsup |aₙ|^{1/n} (Cauchy–Hadamard), never assume the ratio limit exists. Behaviour on the boundary circle is decided separately (Abel); differentiation/integration keep the radius.

Power series — radius, boundary, Abel

The written notes for this page come with the Notes pack. The video, the visual and the practice below are free.

See pricing

See it move

Taylor polynomials and the radius you can seeinteractive

1/(1+x²) is smooth on all of ℝ, yet its series refuses to go past |x| = 1. Watch the wall the poles at ±i put there.

Unlock interactive visuals

The trap here

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

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

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