Question A power series looks like an infinite polynomial: Σ cₙ(x − a)ⁿ. Inside a certain radius it behaves politely — you can differentiate and integrate term by term with care. Outside that radius, it falls apart. Think of a flashlight beam that only stays focused on part of the number line. The radius R marks how far from the center a the series "works." Endpoints of the interval need their own checks — they are not automatic.
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Find the radius of convergence with the ratio (or root) test Test endpoints to get the full interval of convergence State where termwise differentiation/integration is valid Avoid assuming global convergence or ignoring endpoints
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Σ cₙ(x−a)ⁿ infinite polynomial only on part of ℝ Radius R: |x−a| < R works check endpoints!
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