LaTeX
f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^nVariables
- a: expansion point
- f⁽ⁿ⁾: nth derivative
How to use this formula
Represents a sufficiently smooth function as a power series around a point.
Important notes
- Convergence must be checked for the function and interval.
Quick example
For e^x at a=0, all derivatives equal 1.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Convergence must be checked for the function and interval.
- For the Taylor Series, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Taylor Series, the index variable, lower bound, upper bound, and any empty-sum or empty-product convention must be clear.
- The function must satisfy the differentiability, continuity, or integrability conditions required by the operation being used.
Worked example
For e^x at a=0, all derivatives equal 1.
Common mistakes
- When copying Taylor Series, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not apply a differentiation or integration rule outside its domain or omit endpoint, continuity, or constant-of-integration checks.
Continue the workflow
Use Taylor Series in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Calculus Volume 1OpenStax, Rice University — Reviewed limits, derivatives, integrals, and foundational calculus formulas.
Frequently asked questions
What is the Taylor Series used for?
Represents a sufficiently smooth function as a power series around a point.
Can I copy this formula as LaTeX?
Yes. Copy f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.