Taylor Series Approximation Calculator
Approximate sin(x), cos(x), e^x, and ln(1+x) using a Taylor series with 1 to 10 terms.
Shows the approximation vs true value with percent error and a chart.
A Taylor series represents a smooth function as an infinite sum of polynomial terms, each involving a derivative of the function at a chosen center point. The Maclaurin series is the special case centered at x = 0.
Why Taylor series matter. Computers cannot evaluate sin(x) or e^x directly. They use polynomial approximations internally. The Taylor series is the mathematical foundation for those approximations. Engineers use truncated series to simplify equations. Physicists use them constantly for small-angle approximations (sin x ~ x for small x).
The four series computed here (all centered at x = 0):
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + …
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + …
e^x = 1 + x + x^2/2! + x^3/3! + …
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + … (valid for -1 < x <= 1)
Number of terms and accuracy. Adding more terms always improves accuracy near x = 0. Far from the center, you need more terms. For sin(x) at x = 1 radian, 4 terms gives six decimal places of accuracy. At x = 10, you would need 20+ terms.
The percent error is |approximate - true| / |true| x 100. Watch how quickly it drops as you add terms. For alternating series, the error is bounded by the first omitted term.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.