Taylor Series Calculator

Q: What's the difference between Taylor series and Maclaurin series?

A Maclaurin series is a Taylor series centered at a = 0. The general Taylor series uses any center point a, while Maclaurin always expands around zero. Maclaurin series are simpler because (x-a) becomes just x. The series for eˣ, sin(x), and cos(x) are typically written as Maclaurin series since they’re naturally centered at zero.

Q: How do you find the Taylor series of a function?

Calculate successive derivatives at the center point: f(a), f'(a), f”(a), etc. Each term is f⁽ⁿ⁾(a)/n! times (x-a)ⁿ. For sin(x) at a=0: f(0)=0, f'(0)=1, f”(0)=0, f”'(0)=-1, cycling with period 4. This gives x – x³/3! + x⁵/5! – … The pattern in derivatives determines the pattern in coefficients.

Q: What is the radius of convergence?

The radius of convergence R is the distance from the center where the series converges. For |x-a| R, it diverges. At |x-a| = R, you must test individually. Use the ratio test: R = lim|aₙ/aₙ₊₁|. For eˣ, R = ∞ (converges everywhere). For 1/(1-x) centered at 0, R = 1 because of the singularity at x = 1.

Q: Why do calculators use Taylor series?

Computers can only do arithmetic: add, subtract, multiply, divide. Taylor series convert transcendental functions into polynomials that use only these operations. To compute sin(0.5), evaluate the polynomial 0.5 – 0.5³/6 + 0.5⁵/120 – … until you reach desired precision. Modern implementations use optimized variants (Chebyshev, CORDIC) but the core idea is the same.

Q: How many terms do you need for a good approximation?

It depends on how far x is from the center and the required accuracy. Near the center, few terms suffice. For eˣ at x=0.1, three terms give 6 decimal places. At x=2, you need about 10 terms. The error bound R_n helps estimate: if |f⁽ⁿ⁺¹⁾(c)| ≤ M, then |error| ≤ M|x-a|ⁿ⁺¹/(n+1)! Use enough terms until this bound meets your accuracy needs.

Q: What is the small-angle approximation?

For small angles (in radians), sin(θ) ≈ θ and cos(θ) ≈ 1 – θ²/2. These come from keeping just the first non-constant term of each Taylor series. Physics uses these constantly: pendulum period derivation, optics calculations, wave equations. ‘Small’ typically means |θ| < 0.1 radians (about 6°) for good accuracy, though sin(0.2) ≈ 0.2 is still within 1%.

Q: Can every function be represented by a Taylor series?

No. The function must be infinitely differentiable at the center point, and the series must converge to the function. f(x) = e^(-1/x²) (with f(0)=0) has all derivatives equal to zero at x=0, so its Taylor series is just 0—but the function isn’t zero elsewhere. Functions with discontinuities, sharp corners, or singularities can’t have Taylor series at those points.

Q: How are Taylor series related to Euler's formula?

Euler’s formula e^(ix) = cos(x) + i·sin(x) can be derived by substituting ix into the Taylor series for eˣ. The real terms give the cosine series; the imaginary terms give the sine series. This beautiful connection shows that exponential, sine, and cosine are deeply related through complex numbers. It’s one of the most important formulas in mathematics.

Use this free Taylor Series Calculator to expand any function into a Taylor or Maclaurin polynomial around a chosen center point. Enter your function, select the number of terms, and instantly see the series expansion, coefficient table, and interactive graph comparing the original function with its approximation.

Generate Taylor or Maclaurin series expansions for any function.

Function f(x)

Center point (a)

Number of terms

What Is a Taylor Series?

A Taylor series represents a function as an infinite sum of terms calculated from the function’s derivatives at a single point. Named after mathematician Brook Taylor, this powerful tool lets you approximate complex functions using simple polynomials.

The general formula for a Taylor series centered at point $ a $ is:

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f”(a)}{2!}(x-a)^2 + \cdots$$

When the center point $ a = 0 $, the series is called a Maclaurin series.

Common Taylor Series

Exponential Function

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

Converges for all real $ x $.

Sine Function

$$\sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

Converges for all real $ x $.

Cosine Function

$$\cos(x) = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$

Converges for all real $ x $.

Natural Logarithm

$$\ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$$

Converges for $ -1 < x \leq 1 $.

Geometric Series

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots = \sum_{n=0}^{\infty} x^n$$

Converges for $ |x| < 1 $.

Why Taylor Series Matter

In Engineering

Engineers use Taylor approximations to simplify complex calculations in control systems, signal processing, and structural analysis. Small-angle approximations ($ \sin \theta \approx \theta $) come directly from Taylor series.

In Computing

Calculators and computers use Taylor series to compute transcendental functions like sin, cos, and exp. The series provides a way to calculate these values using only addition and multiplication.

Convergence

Not all Taylor series converge everywhere. The radius of convergence determines how far from the center point the series accurately represents the function. For some functions like $ e^x $, the series converges everywhere. For others like $ \ln(1+x) $, it only converges within a limited interval.

Error Estimation

The error in a Taylor polynomial approximation can be bounded using the remainder term:

$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$

where $ c $ is between $ a $ and $ x $.

FAQs

What’s the difference between Taylor series and Maclaurin series?

A Maclaurin series is a Taylor series centered at a = 0. The general Taylor series uses any center point a, while Maclaurin always expands around zero. Maclaurin series are simpler because (x-a) becomes just x. The series for eˣ, sin(x), and cos(x) are typically written as Maclaurin series since they’re naturally centered at zero.

How do you find the Taylor series of a function?

Calculate successive derivatives at the center point: f(a), f'(a), f”(a), etc. Each term is f⁽ⁿ⁾(a)/n! times (x-a)ⁿ. For sin(x) at a=0: f(0)=0, f'(0)=1, f”(0)=0, f”'(0)=-1, cycling with period 4. This gives x – x³/3! + x⁵/5! – … The pattern in derivatives determines the pattern in coefficients.

What is the radius of convergence?

The radius of convergence R is the distance from the center where the series converges. For |x-a| R, it diverges. At |x-a| = R, you must test individually. Use the ratio test: R = lim|aₙ/aₙ₊₁|. For eˣ, R = ∞ (converges everywhere). For 1/(1-x) centered at 0, R = 1 because of the singularity at x = 1.

Why do calculators use Taylor series?

Computers can only do arithmetic: add, subtract, multiply, divide. Taylor series convert transcendental functions into polynomials that use only these operations. To compute sin(0.5), evaluate the polynomial 0.5 – 0.5³/6 + 0.5⁵/120 – … until you reach desired precision. Modern implementations use optimized variants (Chebyshev, CORDIC) but the core idea is the same.

How many terms do you need for a good approximation?

It depends on how far x is from the center and the required accuracy. Near the center, few terms suffice. For eˣ at x=0.1, three terms give 6 decimal places. At x=2, you need about 10 terms. The error bound R_n helps estimate: if |f⁽ⁿ⁺¹⁾(c)| ≤ M, then |error| ≤ M|x-a|ⁿ⁺¹/(n+1)! Use enough terms until this bound meets your accuracy needs.

What is the small-angle approximation?

For small angles (in radians), sin(θ) ≈ θ and cos(θ) ≈ 1 – θ²/2. These come from keeping just the first non-constant term of each Taylor series. Physics uses these constantly: pendulum period derivation, optics calculations, wave equations. ‘Small’ typically means |θ| < 0.1 radians (about 6°) for good accuracy, though sin(0.2) ≈ 0.2 is still within 1%.

Can every function be represented by a Taylor series?

No. The function must be infinitely differentiable at the center point, and the series must converge to the function. f(x) = e^(-1/x²) (with f(0)=0) has all derivatives equal to zero at x=0, so its Taylor series is just 0—but the function isn’t zero elsewhere. Functions with discontinuities, sharp corners, or singularities can’t have Taylor series at those points.

How are Taylor series related to Euler’s formula?

Euler’s formula e^(ix) = cos(x) + i·sin(x) can be derived by substituting ix into the Taylor series for eˣ. The real terms give the cosine series; the imaginary terms give the sine series. This beautiful connection shows that exponential, sine, and cosine are deeply related through complex numbers. It’s one of the most important formulas in mathematics.