How to use the taylor series calculator
- Choose a function. For a power of x, also enter the power p.
- Enter the center a, the order N (the highest power) and the point x where you want to evaluate it.
- Read the polynomial, its value at x, the true value, the actual error and the Lagrange error bound.
- Use the table of orders to see how the error shrinks as terms are added.
Formula
T_N(x) = sum for n = 0 to N of f^(n)(a) / n! x (x - a)^n
e^x: c_n = e^a / n!; sin x: c_n = sin(a + n pi / 2) / n!; cos x: c_n = cos(a + n pi / 2) / n!
ln x: c_n = (-1)^(n+1) / (n a^n); 1 / x: c_n = (-1)^n / a^(n+1)
x^p: c_n = C(p, n) a^(p - n)
Remainder: |f(x) - T_N(x)| <= M |x - a|^(N+1) / (N+1)!, M = largest |f^(N+1)| between a and x
C(p, n) is the generalized binomial coefficient p (p - 1) ... (p - n + 1) / n!. For center 0 and the functions e^x, sin, cos, sinh and cosh the polynomial is shown with exact fractions.
sin x around 0 and ln x around 1
For sin x with center 0 and order 5 the polynomial is x - x^3 / 6 + x^5 / 120. At x = 1 it gives 0.8416666667 against the true 0.8414709848, an error of 0.0001957. The Lagrange bound is 1 / 720 = 0.001389, so the error is within the bound.
For ln x with center 1 and order 5 the coefficients are 0, 1, -1/2, 1/3, -1/4 and 1/5. At x = 1.5 the polynomial gives 0.4072916667 against ln 1.5 = 0.4054651081. The bound is 0.002604.
What a Taylor polynomial does
A Taylor polynomial replaces a smooth function by a polynomial that matches its value and its first N derivatives at the center. Near the center the two are very close, and the polynomial is easy to compute by hand or in code. Calculators and computers use this idea, in various forms, to evaluate functions such as sine and exponentials.
Adding terms usually improves the approximation near the center. The table of orders shows the error for each order at your point, so you can see how many terms are enough for a target accuracy.
Error and where the series works
The Lagrange remainder gives a guaranteed upper bound on the error, using the largest size of the next derivative between the center and the point. The bound is often larger than the actual error, but it never underestimates it. For e^x, sin x, cos x, sinh x and cosh x the series converges for every x. For ln x, 1 / x and powers it converges only within a distance of a from the center, so points far from the center give poor results at any order.
For roots of polynomials try the polynomial root finder, and for the algebra of polynomials see the polynomial arithmetic calculator. The logarithm, exponent and root calculator evaluates the true values these polynomials approximate.
Assumptions and limits
- The function is one of the listed ones; custom expressions are not supported because derivatives are taken from closed forms.
- For ln x and powers of x the center and the point must be above 0. For 1 / x they must be on the same side of 0.
- Angles for sine and cosine are in radians.
- Order is limited to 20, and values are shown with up to 10 significant digits.
Frequently asked questions
What is the difference between a Taylor and a Maclaurin series?
A Maclaurin series is a Taylor series with the center at 0. Setting the center to 0 here gives the Maclaurin polynomial of the chosen function.
Why does the error bound exceed the actual error?
The bound uses the worst case of the next derivative anywhere between the center and the point. The real error depends on the derivative at one unknown point in that range, so it is often smaller.
Why is the result poor for ln x far from the center?
The series for ln x around 1 converges only for x between 0 and 2. Outside that range adding terms makes the polynomial worse, not better, so a different center is needed.
Can I use degrees for sine and cosine?
The page works in radians, as Taylor series do. To use a value in degrees, multiply it by pi and divide by 180 first, and enter the result as x.