Mathematics · University
Taylor series, convergence and error · revision notes
The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
Key ideas and conditions
Coefficients
The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
Radius
The radius of convergence controls an open interval or disk; endpoints require separate tests.
Remainder
A bound on the next derivative over the relevant interval yields a Taylor remainder bound.
Alternation
An alternating-series error bound requires terms decreasing in magnitude to zero.
Explain without looking
- For the alternating series 1−1/2+1/3−1/4+..., what upper bound on absolute error follows after the fourth term?
- Use a geometric-series identity to derive 1/(1−x)=sum x^n for abs(x)<1.
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.