Differentiating and Integrating Power Series

Real Analysis · Axiom Academy

LESSON Differentiating and Integrating Power Series Inside the radius of convergence a power series behaves like an infinite polynomial — you can differentiate and integrate it one term at a time, and the radius never changes. Take a power series centered at a with radius . On the open interval it is differentiable, and you differentiate it exactly as you would a polynomial: power rule on each term. The n th coefficient c_n gets multiplied by n and the power drops by one. Theorem (Term-by-term differentiation) then f is differentiable on and with the same radius of convergence R . Example — the geometric series Differentiating term by term collapses the constant term and rescales the rest: Integration runs the operation in reverse: each power goes up by one and you divide by the new power n+1 . Antidifferentiating a known series is how you discover the series of a brand-new function — watch the partial sums of pile up onto the true curve. Theorem (Term-by-term integration) again valid for , same radius R . Start from the geometric series (valid for ): Integrate from 0 to x term by term: 3. Application — the Series for The same machinery delivers one of the most famous series in analysis. Start from a geometric series, substitute to land on , then integrate term by term. 4. Same Radius R , Endpoints May Flip

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