Taylor Series Visualizer

What is a Taylor series?

The Taylor polynomial of degree \(n\) of a function \(f\) about a point \(a\) is the unique polynomial of degree \(\le n\) that matches \(f\) and its first \(n\) derivatives at \(a\). It is the best local polynomial model of \(f\) near \(x=a\):

When the expansion point is \(a=0\) this is called the Maclaurin polynomial, and the powers \((x-a)^k\) become simply \(x^k\). As \(n\to\infty\) (inside the radius of convergence) the polynomials approach \(f\) — this limit is the Taylor series.

Taylor’s theorem and the remainder

The error \(R_n(x)=f(x)-T_n(x)\) is controlled by the Lagrange remainder \(R_n(x)=\dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}\) for some \(\xi\) between \(a\) and \(x\). The approximation is excellent near \(a\) and generally degrades as \(|x-a|\) grows; outside the radius of convergence the polynomials may diverge from \(f\) no matter how high the degree (try \(\tfrac{1}{1-x}\) or \(\tfrac{1}{1+x^2}\)).

How the coefficients are computed

The derivatives \(f^{(k)}(a)\) are obtained by exact symbolic differentiation of the input expression (repeatedly applying the differentiation rules), then evaluated at \(a\). If the expression cannot be differentiated symbolically, the tool falls back to high-order central finite differences — the same numerical-differentiation idea used in the Finite Difference Generator. Each coefficient \(\dfrac{f^{(k)}(a)}{k!}\) is then reduced to a simplified fraction whenever the value is rational (e.g. \(\tfrac{1}{2},\tfrac{1}{6},\tfrac{1}{120},-\tfrac{1}{3}\)); otherwise a decimal is shown.

How to use this tool

• Enter a function \(f(x)\) (e.g. sin(x), exp(x), 1/(1+x^2)) and an expansion point \(a\) (math expressions like pi/2 are accepted).
• Set the degree \(n\), then click Plot Taylor Series.
• Press Add next degree (+1) repeatedly to increase \(n\) one step at a time and watch the polynomial hug the curve over a wider interval.
• The expansion point \((a,f(a))\) is marked on the graph with a dot and a vertical guide line.
• Pick a ready-made case from the Examples drop-down menu.
• Adjust the plot window \([x_{\min},x_{\max}]\) to zoom in or out; download the figure as a PNG.