Taylor Series Approximation

Approximate sin/cos/eˣ/ln(1+x) via Taylor polynomial of degree n.

Inputs

Result

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How to use this calculator

  • Pick function, x, and polynomial order.

About this calculator

Taylor series approximates a function as a polynomial around a point. Maclaurin: around 0. sin(x) ≈ x − x³/6 + x⁵/120 − …; cos(x) ≈ 1 − x²/2 + x⁴/24 − …; eˣ ≈ 1 + x + x²/2 + x³/6 + …. More terms = better approximation. Used in calculator firmware (chip can't natively compute trig), physics (small-angle approx sin x ≈ x), and numerical analysis. Source: NIST DLMF Ch. 1.

Frequently asked

Why polynomial?+
Polynomials are easy to compute (just multiplications + additions). Most functions can be approximated by them in some range.
Convergence radius?+
sin/cos/exp: converges for all x. ln(1+x): only |x| < 1. Beyond radius: series diverges (use different expansion point).
Why factorial in denominator?+
Comes from differentiating xⁿ multiple times: nth derivative gives n! coefficient.
How many terms?+
For ε accuracy: depends on x and function. Smaller x: fewer terms. For x near 1, need 10-15 terms for 6 decimals.
Real-world use?+
Calculator chips compute sin/cos via Taylor. Physics: small-angle approximations. Numerical methods.

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