Taylor Series Approximation
Approximate functions with Taylor polynomials
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What the defaults give
Six terms of the sine series at x = 1 give 0.841471.
The true value of sin(1) is also 0.841471, so the approximation is exact to six decimal places.
The formulas
sin x = x − x³/3! + x⁵/5! − x⁷/7! + …
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …
eˣ = 1 + x + x²/2! + x³/3! + …
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + …
Each term you add is the next correction in the polynomial. The factorial in the denominator is what makes the terms shrink quickly for small x, which is why a handful of terms is enough near zero and not nearly enough far from it.
Worked examples
sin(1), 6 terms → 0.841471 (true 0.841471)
cos(1), 6 terms → 0.540302 (true 0.540302)
e¹, 6 terms → 2.716667 (true 2.718282, short by 0.0016)
sin(2), 4 terms → 0.907937 (true 0.909297)
ln(1.5), 6 terms → 0.407292 (true 0.405465)
sin(3), 3 terms → 0.525000 (true 0.141120)
Radius of convergence is the whole story
The sine and cosine series converge for every x, but slowly for large x. The last example shows what happens when you truncate early: at x = 3 with only three terms the approximation is 0.525 against a true 0.141, an error larger than the answer. Adding terms fixes it — the series is convergent, just not yet converged.
For the logarithm the news is worse. ln(1+x) only converges for x above −1 and up to 1, because the series is built around x = 0 and the function has a singularity at x = −1. Feeding x = 2 gives a series that diverges no matter how many terms you add. Every Taylor series has such a boundary, and outside it more terms make things worse, not better.
Why anyone uses this
Because a polynomial can be evaluated with add and multiply, which is all a processor does cheaply. Calculators and libraries compute sin, cos and eˣ exactly this way, with range reduction first — folding the argument into a small interval where a short series is accurate — and then a polynomial fitted to that interval.
The same idea underlies physics approximations: sin θ ≈ θ for small angles, and (1 + x)ⁿ ≈ 1 + nx for small x. Those are first-order Taylor expansions, and they are accurate enough that the error is below measurement over the range where they are used. The skill is knowing the range, which is what the terms and x fields let you test.
Frequently Asked Questions
How accurate is a 6-term Taylor series for sin(1)?
Exact to six decimal places: 0.841471, the same as the true value. Sine converges quickly for small arguments.
Why does sin(3) with 3 terms give the wrong answer?
The series has not converged yet. Three terms give 0.525 against a true 0.141; adding terms brings it down toward the correct value.
What is the radius of convergence?
The range of x where the series converges. Sine and cosine converge everywhere, eˣ everywhere, and ln(1+x) only for x between −1 and 1.
What does the ln option actually compute?
The series for ln(1+x), so entering 0.5 approximates ln(1.5) = 0.4055. Entering 2 asks for ln(3) but the series diverges there.
Where is this used in practice?
In calculators and math libraries, which evaluate polynomials after range reduction, and in physics approximations such as sin θ ≈ θ for small angles.
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