Taylor’s Theorem with Remainder
Lagrange form of the error term.
Key idea
Lagrange form of the error term.
In depth
f(x) = Σ f⁽ᵏ⁾(a)(x − a)ᵏ/k! + Rₙ with Rₙ = f⁽ⁿ⁺¹⁾(c)(x − a)ⁿ⁺¹/(n + 1)!. The remainder shows how well the polynomial approximates f and decides if the Taylor series converges to f.
Worked example
For eˣ on [0, 1], Rₙ ≤ e/(n + 1)! → 0, so the series converges to eˣ.
Real-world uses
- Bounding the error of approximations used in calculators and sensor calibration
Common mistake
Assuming the Taylor series always converges to the function.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Taylor’s Theorem with Remainder?
Lagrange form of the error term. It is a topic at the expert level in Analysis & Topology.
Where is Taylor’s Theorem with Remainder used in real life?
Real-world examples: Bounding the error of approximations used in calculators and sensor calibration.
What should I learn before Taylor’s Theorem with Remainder?
Make sure you are comfortable with Power Series Radius of Convergence, the previous topic in Analysis & Topology.
What is a common mistake with Taylor’s Theorem with Remainder?
Assuming the Taylor series always converges to the function.
How can I practise Taylor’s Theorem with Remainder?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Taylor’s Theorem with Remainder?
Move on to Analytic Functions next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.