QiLing Tools

∑Taylor Series Calculator - Taylor & Maclaurin Polynomials with Error and Graph

A Taylor expansion writes a function near a point as a polynomial: f(x) ≈ f(a) + f′(a)(x−a) + f″(a)/2!·(x−a)² + …. Enter the function and the centre and you get the Taylor polynomial of order n, every coefficient (as a fraction such as 1/6 or −1/5040 where possible), the true error at a point of your choice, and a plot of the function together with several orders so you can watch the approximation improve. Centre 0 gives the Maclaurin series.

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How to use Taylor Series

  1. 1Enter f(x), for example sin(x), e^x, ln(1+x) or 1/(1-x)
  2. 2Enter the centre a (expressions like pi/4 work) and drag to choose the order n
  3. 3Read the Taylor polynomial, the coefficient table and the radius estimate
  4. 4Enter a check point x to see the error, pick orders to compare on the plot, or open it in the grapher

FAQ

What is the difference between a Taylor and a Maclaurin series?
A Maclaurin series is simply a Taylor series centred at a = 0, e.g. e^x = 1 + x + x²/2! + x³/3! + … and sin x = x − x³/3! + x⁵/5! − …. Enter 0 as the centre to get it.
How are the coefficients computed, and are they exact?
x is written as a + t and every operation in the expression — arithmetic, powers, trig, exponentials, logarithms — is carried out on power series in t. This is forward-mode automatic differentiation: the derivatives come from exact recurrences with only floating-point rounding, unlike finite differences, which fall apart at high orders.
Why does a higher order make the error worse far from the centre?
A Taylor series only converges inside its radius of convergence. For 1/(1−x) or ln(1+x) around 0 the radius is 1, so at x = 1.5 adding terms makes things worse, not better. The tool estimates the radius and warns you when your check point is outside it.
Which functions cannot be expanded?
Functions that are undefined or not differentiable at the centre, such as ln x and √x at 0, |x| at 0 or tan x at π/2. Pick another centre instead — ln x expands nicely around a = 1.