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.