Taylor Series

By SkytrailGroup Mathematics Editorial Team

Taylor Series

\[f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n\]

Common Series (a=0)

PropertyStatement
\(e^x\)\(\sum x^n/n!\)
\(\sin x\)\(\sum(-1)^n x^{{2n+1}}/(2n+1)!\)
\(\cos x\)\(\sum(-1)^n x^{{2n}}/(2n)!\)
\(\ln(1+x)\)\(\sum(-1)^{{n+1}}x^n/n\), |x|≤1

Examples

Example 1. Approximate \(e^{{0.1}}\) using 3 terms.

Solution. \(1+0.1+0.01/2=1.105\). Exact: ≈1.10517.

Deep Dive: Taylor Series

This section builds durable understanding of taylor series in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.

Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.

Visual Intuition

Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.

Checklist: domain constraints - symmetry - limiting behavior - sanity check at special values.

Practice Set

Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.

Target. Your final line should include assumptions, derivation path, and a quick verification.

Practice B. Build an application scenario using taylor series and solve it with both symbolic and numeric methods.

Target. Compare outputs and explain any approximation gap.

References & Editorial Notes

  • Stewart, Calculus.
  • Strang, Introduction to Linear Algebra.
  • Apostol, Mathematical Analysis.

Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.