Taylor Series Calculator | DigitHelm
Compute Taylor/Maclaurin series expansions for common functions.
Common Expansions
What Is the Taylor Series Calculator | DigitHelm?
The Taylor Series Calculator computes the Taylor polynomial P_n(x) of any function f(x) centred at a point a, up to order n (maximum 10). For each term, it numerically computes the nth derivative f⁽ⁿ⁾(a) using high-order central difference formulas, then builds the coefficient cₙ = f⁽ⁿ⁾(a)/n!. Enter a test x to compare P_n(x) against the true f(x) and see the approximation error. A graph overlays the polynomial and the original function.
- ›Coefficients: cₙ = f⁽ⁿ⁾(a)/n! computed via high-order numerical differentiation
- ›Maclaurin series is the special case a = 0
- ›Error |f(x) − P_n(x)| decreases as n increases (within the radius of convergence)
- ›Term table shows n, f⁽ⁿ⁾(a), n!, cₙ, and the symbolic term expression
- ›Graph shows original f(x) (solid) vs polynomial approximation (dashed)
Formula
Taylor Series Formula
Taylor Series
f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
Maclaurin
a = 0: f(x) = Σ f⁽ⁿ⁾(0)/n! · xⁿ
eˣ
1 + x + x²/2! + x³/3! + … (all ℝ)
sin(x)
x − x³/3! + x⁵/5! − … (all ℝ)
cos(x)
1 − x²/2! + x⁴/4! − … (all ℝ)
Remainder Rₙ
|Rₙ| ≤ M|x−a|ⁿ⁺¹/(n+1)! (Taylor)
How to Use
- 1Enter f(x) using standard function notation (exp(x), sin(x), ln(x), x^3, etc.)
- 2Set the expansion centre a (use a=0 for Maclaurin series)
- 3Set the polynomial order (1–10); higher orders give better approximations
- 4Enter a test x value to evaluate and compare f(x) vs P_n(x)
- 5Click Expand Series to see coefficients, the polynomial, and the error
- 6Use presets (eˣ, sin(x), cos(x), ln(1+x), 1/(1−x)) for instant examples
Example Calculation
Maclaurin series for eˣ up to order 4, evaluate at x=1:
P₄(x) = 1 + x + x²/2 + x³/6 + x⁴/24
P₄(1) = 1+1+0.5+0.1667+0.04167 = 2.7083
e¹ = 2.71828...
Error = |2.71828 − 2.7083| = 0.00998
Taylor series for sin(x) at a=0, order 5:
At x=π/4 ≈ 0.7854:
P₅ = 0.7854 − 0.0807 + 0.00024 = 0.7071
sin(π/4) = 0.70711... Error < 10⁻⁵
Radius of Convergence
Every Taylor series has a radius of convergence R: the series equals f(x) for |x−a| < R and diverges for |x−a| > R. For eˣ, sin(x), cos(x): R = ∞ (converges everywhere). For ln(1+x) at a=0: R = 1 (converges for −1 < x ≤ 1). For 1/(1−x) at a=0: R = 1 (geometric series). Using the polynomial outside its radius of convergence gives wildly incorrect results, the calculator shows the actual error so you can see this directly.
Understanding Taylor Series | DigitHelm
Key Taylor and Maclaurin Series
| Function | Series (a=0) | First 3 Terms | Radius R |
|---|---|---|---|
| eˣ | Σ xⁿ/n! | 1 + x + x²/2 | ∞ |
| sin(x) | Σ (−1)ⁿx²ⁿ⁺¹/(2n+1)! | x − x³/6 + x⁵/120 | ∞ |
| cos(x) | Σ (−1)ⁿx²ⁿ/(2n)! | 1 − x²/2 + x⁴/24 | ∞ |
| ln(1+x) | Σ (−1)ⁿ⁺¹xⁿ/n | x − x²/2 + x³/3 | 1 |
| 1/(1−x) | Σ xⁿ | 1 + x + x² | 1 |
| arctan(x) | Σ (−1)ⁿx²ⁿ⁺¹/(2n+1) | x − x³/3 + x⁵/5 | 1 |
| (1+x)ᵅ | Σ C(α,n)xⁿ | 1 + αx + α(α−1)x²/2 | 1 |
| sinh(x) | Σ x²ⁿ⁺¹/(2n+1)! | x + x³/6 + x⁵/120 | ∞ |
Frequently Asked Questions
What is a Taylor series?
Discovered by Brook Taylor (1715) and used by Newton, Leibniz, and Euler. The Maclaurin series (a=0) is named after Colin Maclaurin who popularized it.
- ›f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + ...
- ›Each term adds a higher-degree correction to the approximation
- ›First two terms = tangent line approximation
- ›The series converges to f(x) within the radius of convergence
What is the Maclaurin series?
- ›eˣ: all terms positive, converges for all x (R = ∞)
- ›sin(x): only odd powers, alternating signs (R = ∞)
- ›cos(x): only even powers, alternating signs (R = ∞)
- ›1/(1−x) = 1+x+x²+... (geometric; converges only |x| < 1)
- ›ln(1+x): converges only for −1 < x ≤ 1
How is the approximation error calculated?
- ›Error shrinks as n increases (more terms = better approximation)
- ›Error grows as |x−a| increases (further from centre = worse)
- ›For eˣ at x=1 with n=4: error ≈ 0.010 (3 decimal places correct)
- ›With n=7: error ≈ 3×10⁻⁵ (5 decimal places correct)
What is the radius of convergence?
- ›R = ∞: eˣ, sin(x), cos(x), entire real line
- ›R = 1: ln(1+x), 1/(1−x), arctan(x)
- ›At the boundary x = a ± R: may converge or diverge (test individually)
- ›Computed by ratio test: R = lim |cₙ/cₙ₊₁| as n → ∞
What are Taylor series used for?
- ›Computer arithmetic: sin/cos/exp computed via polynomial approximations
- ›Physics: (1−v²/c²)^(−1/2) ≈ 1+v²/2c² for v << c (relativistic kinetic energy)
- ›Small-angle approximation: sin(θ) ≈ θ for the pendulum period formula
- ›Numerical analysis: finite-difference formulas derived from Taylor expansion
How is the nth derivative computed numerically?
- ›f'(a) ≈ [f(a+h)−f(a−h)]/(2h), second-order accurate
- ›f''(a) ≈ [f(a+h)−2f(a)+f(a−h)]/h², second-order accurate
- ›f'''(a) uses 5-point stencil for better accuracy
- ›Step size h balances truncation error (large h) vs rounding error (small h)
Is this Taylor series calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser.