← MATH 01131 · Calculus II
MATH 01131 · Sequences & series
Power series / Taylor & Maclaurin basics
Definition
Power series Σ cₙ(x−a)ⁿ represents functions on an interval of convergence. Taylor series uses cₙ=fⁿ(a)/n!; Maclaurin is Taylor at 0.
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Memorize eˣ, sin x, cos x, 1/(1−x) Maclaurin cores and manipulate (differentiate, integrate, substitute).
Remainder/Lagrange error bounds appear when approximating.
A function can equal its Taylor series on an interval — or not (smooth but non-analytic pathologies exist; calc courses focus on standard functions).
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Attempt each problem first, then open the worked solution.
1.Write Maclaurin polynomials for eˣ through degree 3.