Mr. Integral
Math notes/Calculus II/Power Series and Taylor Approximation

Taylor Polynomials and Remainders — Cheat sheet

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Core rule

Pn(x)=∑k=0nf(k)(a)k!(x−a)k,∣Rn(x)∣≤M∣x−a∣n+1(n+1)!.P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k,\quad |R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

Watch for

Bound the derivative on the whole interval between center and input. Smoothness alone does not imply equality with a Taylor series.