Taylor Series Calculator
Taylor and Maclaurin expansions of sin, cos, eˣ, ln(1+x) and 1/(1−x) to any order up to 15 — every coefficient computed from the derivatives, and the truncation error measured against the true value.
Example — Taylor series of sin x around 0, to order 5x − x³/6 + x⁵/120
How we got it — the example, step by step
- Coefficients are f⁽ᵏ⁾(0)/k!: the derivatives of sin cycle sin, cos, −sin, −cos, so at 0 they give 0, 1, 0, −1, 0, 1…
- Divide by the factorials: 1/1! = 1, −1/3! = −1/6, 1/5! = 1/120. The even terms vanish.
Check it backwards
At x = 0.5 the series gives 0.4794271. sin(0.5) itself is 0.4794255. The gap, about 1.5 × 10⁻⁶, is the truncation error — and the page shows it rather than hiding it, because that gap is the honest price of stopping at order 5.
Questions people also ask
Why does the ln(1+x) series refuse x = 2?
Because it diverges there: the series for ln(1+x) converges only for −1 < x ≤ 1. Outside the radius of convergence, more terms make the sum worse, not better — so the tool says so instead of printing a confident wrong number.