Taylor Series Calculator – Expand Functions

A Taylor series rebuilds a function from its derivatives at a single point. Expanding eˣ about 0 (a Maclaurin series): every derivative is 1, so eˣ = 1 + x + x²/2! + x³/3! + … At x = 1 the partial sum 1+1+0.5 = 2.5 underestimates e ≈ 2.71828 by 0.218; adding x³/6 (≈0.1667) lands at 2.66667, and the series closes in factorially fast. For sin x the odd series x − x³/6 + x⁵/120 gives sin(0.3) ≈ 0.2955 versus the true 0.29552 — five-digit accuracy from two terms because the next term 0.3⁵/120 ≈ 2×10⁻⁵ bounds the error (alternating series). The calculator expands the common library (eˣ, sin, cos, ln(1+x), 1/(1−x)) about any center, prints term-by-term values, running partial sums, and the truncation error estimate via the next term. Radius of convergence is flagged — ln(1+x) only converges for −1 < x ≤ 1.

1. 1

   Choose a function and center

   eˣ about 0, or sin x about π/6 for example

2. 2

   Pick the order

   How many terms to keep in the polynomial

3. 3

   Evaluate term by term

   Each (x−a)ᵏ/factorial term and partial sum is listed

4. 4

   Bound the error

   The first omitted term estimates the truncation error

• 1

  Maclaurin = Taylor centered at 0

• 2

  Factorial denominators make convergence fast for eˣ and sin

• 3

  The first dropped term bounds the error for alternating series

• Forgetting the k! divisor on each term

• Evaluating (x−a)ᵏ as xᵏ when the center a ≠ 0

• Ignoring the radius of convergence for ln(1+x)

• Adding terms with mismatched signs in the sin series