Power Series Manipulations

Calculus 2 · Axiom Academy

LESSON Power Series Manipulations A power series is a polynomial you can bend — add, shift, substitute, differentiate, and integrate it term by term to build brand-new series from one you already know. Every manipulation below starts from a single known expansion — the geometric series . On its interval , adding more terms (the partial sums ) homes in on the true curve . That convergence is what lets us treat the series as a polynomial we can bend. 2. Add and Subtract, Term by Term Two series can be combined by lining up matching powers and adding their coefficients. Take the expansions of e^ x and e^ -x : average them, and every odd -power term cancels while the even -power terms survive — out drops the series for . When you add or subtract two series, the result converges at least wherever both originals do — its interval is the intersection of the two. Here both series converge for all x , so does too. 3. Multiply by x^k and Substitute Two cheap moves reshape a series fast. Multiplying by x^k slides every exponent up by k . Substituting swaps the variable: flips alternating signs on, and doubles every exponent (and rescales the radius). — same radius, exponents shifted up. A convergent series may be rearranged and re-indexed freely inside its interval — substitution just renames x .

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