Taylor Series Calculator
The ultimate Taylor and Maclaurin series polynomial calculator for computing power series expansions, step-by-step derivative tables, radius of convergence, and real-time approximation curves.
f(x) vs Taylor Polynomial Pₙ(x) Curve
Interactive Approximation PlotStep-by-Step Derivative Table & Factorial Coefficients Formula: f⁽ᵏ⁾(a) / k! × (x − a)ᵏ
| k (Order) | k-th Derivative f⁽ᵏ⁾(x) | Evaluated f⁽ᵏ⁾(a) | k! (Factorial) | Term Coefficient |
|---|
How to Calculate a Taylor Series Polynomial
A Taylor series represents a smooth function f(x) as an infinite polynomial centered at a point a: f(x) = f(a) + f'(a)(x - a) + [f''(a)/2!](x - a)² + ... + [f^(n)(a)/n!](x - a)^n. When centered at a = 0, it is called a Maclaurin series. The degree-n Taylor polynomial P_n(x) approximates f(x) locally around a with error bounded by Taylor's Remainder Theorem.
Anatomy of Taylor & Maclaurin Series Expansions
The Taylor series is one of the most powerful foundational tools in mathematical analysis. It transforms complicated non-linear functions (such as exponentials, logarithms, and trigonometric ratios) into simple polynomials made purely of basic addition and multiplication.
By matching the function value and all successive derivatives at a chosen point x = a, the Taylor polynomial mimics the local curvature, concavity, and higher-order inflection of the original curve.
The anchor point where the approximation is exact with zero error.
Compensates for the power rule during repeated differentiation.
The special case centered at the origin, yielding powers of xᵏ.
Derivation of the Taylor Polynomial Formula & Factorial Division
Suppose we want to approximate a smooth function f(x) near x = a using a general polynomial:
Differentiating repeatedly and evaluating at x = a:
- Zero-th derivative: P(a) = c₀ = f(a) ⇒ c₀ = f(a) / 0!
- First derivative: P'(a) = c₁ = f'(a) ⇒ c₁ = f'(a) / 1!
- Second derivative: P''(a) = 2·1·c₂ = f''(a) ⇒ c₂ = f''(a) / 2!
- k-th derivative: P⁽ᵏ⁾(a) = k!·cₖ = f⁽ᵏ⁾(a) ⇒ cₖ = f⁽ᵏ⁾(a) / k!
Radius & Interval of Convergence (Ratio Test)
An infinite Taylor series does not automatically converge to f(x) for every value of x. The set of all x values where the series converges is called the Interval of Convergence I, and half its width is the Radius of Convergence R.
Functions like eˣ, sin(x), cos(x), sinh(x), cosh(x) have derivatives that grow slower than k!, making their series valid for all real and complex numbers (−∞, ∞).
Functions with singularities like 1/(1 − x) (vertical asymptote at x = 1) or ln(1 + x) converge only within |x| < 1.
Taylor's Theorem & Lagrange Error Bound Estimation
When using a degree-n Taylor polynomial in engineering or physics, we need a mathematical guarantee on the maximum possible error. Taylor's Remainder Theorem (Lagrange Form) provides this exact bound:
Where M is the maximum value of |f⁽ⁿ⁺¹⁾(t)| for any point t between the center a and evaluation point x.
Step-by-Step Worked Examples (eˣ, sin(x), ln(1+x))
Find the 5th-order Maclaurin series for f(x) = sin(x) centered at a = 0.
1. f(0) = sin(0) = 0 → Term 0 = 0.
2. f'(0) = cos(0) = 1 → Term 1 = (1/1!)x = x.
3. f''(0) = -sin(0) = 0 → Term 2 = 0.
4. f'''(0) = -cos(0) = -1 → Term 3 = (-1/3!)x³ = -x³/6.
5. f⁽⁴⁾(0) = sin(0) = 0 → Term 4 = 0.
6. f⁽⁵⁾(0) = cos(0) = 1 → Term 5 = (1/5!)x⁵ = +x⁵/120.
Result: P₅(x) = x − x³/6 + x⁵/120.
Common Calculation Pitfalls & Point of Expansion Errors
When expanding around a ≠ 0 (e.g. a = 1), writing xᵏ instead of (x − 1)ᵏ invalidates the approximation.
Using the series for ln(1 + x) at x = 3 causes the polynomial to explode towards infinity instead of converging.
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.