Taylor Series Approximation – Generate Taylor Polynomials

Generate Taylor series approximations for common functions with our free online calculator. Get polynomial expansions, coefficients, and error analysis step by step.

Function:

Center (a)

Expansion point

Order (n)

Degree of polynomial

Evaluate at x

Point to approximate

Examples:

Understanding Taylor Series

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It's like expressing a complex curve as a polynomial – and polynomials are much easier to work with. The more terms you include, the better the approximation.

Taylor series are the backbone of computational mathematics. Your calculator uses them to compute sin, cos, eˣ, and ln. Physics simulations use them to approximate complex forces. Engineers use them to linearize nonlinear systems. They turn the impossible into the manageable.

Common Taylor Series

Sine (centered at 0)

sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...

Only odd powers, alternating signs

Cosine (centered at 0)

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Only even powers, alternating signs

Exponential (centered at 0)

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

All powers, all positive coefficients

Natural Log (centered at 1)

ln(x) = (x-1) - (x-1)²/2 + (x-1)³/3 - ...

Valid for 0 < x ≤ 2

Worked Examples

Example 1: sin(x) approximation

Problem: Approximate sin(0.5) using Taylor polynomial of order 5 at a=0

T₅(x) = x - x³/6 + x⁵/120

T₅(0.5) = 0.5 - 0.125/6 + 0.03125/120 = 0.5 - 0.02083 + 0.00026

T₅(0.5) ≈ 0.47943 (actual: 0.47943, error: 0.000002%)

Example 2: eˣ approximation

Problem: Approximate e^0.5 using Taylor polynomial of order 4 at a=0

T₄(x) = 1 + x + x²/2 + x³/6 + x⁴/24

T₄(0.5) = 1 + 0.5 + 0.125 + 0.02083 + 0.00260

T₄(0.5) ≈ 1.64844 (actual: 1.64872, error: 0.017%)

Example 3: cos(x) approximation

Problem: Approximate cos(1) using Taylor polynomial of order 6 at a=0

T₆(x) = 1 - x²/2 + x⁴/24 - x⁶/720

T₆(1) = 1 - 0.5 + 0.04167 - 0.00139

T₆(1) ≈ 0.54028 (actual: 0.54030, error: 0.004%)

Example 4: Why center matters

Problem: Approximate sin(3) using order 5 at a=0 vs a=π

At a=0: x=3 is far from center, poor approximation

At a=π: x=3 is close to π≈3.14, excellent approximation. Always center near your evaluation point!

Example 5: Order vs accuracy

Problem: How does order affect e^1 approximation?

Order 2: 1 + 1 + 0.5 = 2.5 (error: 8%)

Order 4: 2.7083 (error: 0.15%)

Order 10: 2.71828 (error: 0.00001%). Higher order = better accuracy.

Quick Fact

Brook Taylor published his method in 1715, but the Scottish mathematician James Gregory discovered many special cases 40 years earlier. The Taylor series for sin(x), cos(x), and eˣ were known to Indian mathematicians of the Kerala school in the 14th century – 300 years before Taylor! Mathematics is a global, cumulative enterprise.

When Taylor Series Work Best

Near the Center

Taylor series converge fastest near the expansion point a. Error grows as you move away. For best results, center at or near your evaluation point.

Smooth Functions

Functions must be infinitely differentiable at the center. Sharp corners, discontinuities, or vertical tangents break the approximation.

Higher Order

More terms = better accuracy. But diminishing returns set in. Order 5-10 often gives excellent results for common functions.

Frequently Asked Questions

What's the difference between Taylor and Maclaurin series?

A Maclaurin series is just a Taylor series centered at a=0. All Maclaurin series are Taylor series, but not vice versa. The term "Maclaurin" honors Colin Maclaurin, who extensively used these 0-centered expansions.

How many terms do I need?

It depends on desired accuracy and distance from center. For sin(x) near 0, order 5-7 gives calculator precision. For eˣ, you might need order 10-15. Use the error term to estimate: |Rₙ| ≤ M|x-a|ⁿ⁺¹/(n+1)!

Do Taylor series always converge?

No! Some functions have Taylor series that converge only within a certain radius. For ln(x) centered at 1, the series only converges for 0 < x ≤ 2. Outside this range, adding more terms makes things worse.

Why are factorials in the denominator?

The n! comes from repeatedly differentiating (x-a)ⁿ. Each differentiation brings down a power: d/dx(xⁿ) = nxⁿ⁻¹, then n(n-1)xⁿ⁻², etc. After n derivatives, you get n!. The factorial normalizes the coefficients.

Can I use Taylor series for any function?

The function must be "analytic" – infinitely differentiable with a convergent Taylor series. Most common functions (polynomials, trig, exp, log) are analytic where defined. Functions with discontinuities or sharp corners are not.

How do calculators use Taylor series?

Calculators use optimized polynomial approximations (often Taylor-based) combined with range reduction. For sin(x), they reduce x to [0, π/2], then use a carefully chosen polynomial. This is faster than computing infinite series directly.

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