STEM Calculators

Taylor Series Generator

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published Updated
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1. 1. Taylor Series Generator
Expands \(f(x)\) into a Taylor series about \(x=a\) (Maclaurin when \(a=0\)). Computes coefficients \(c_k=\dfrac{f^{(k)}(a)}{k!}\), shows a remainder term, and plots partial sums.
Inputs
Use variable x. Constants: pi, e. Supported: + − * / ^, parentheses, sin cos tan, asin acos atan, ln (or log), sqrt, abs, exp. Implicit multiplication like 2x is allowed.
Taylor about \(x=a\). Maclaurin when \(a=0\).
Computes terms through degree \(N\) and shows \(O((x-a)^{N+1})\).
Shows \(f(x_0)\), \(P_N(x_0)\), and \(|f-P_N|\). Also a simple remainder bound.
Click a preset to load & evaluate.
Ready
Graph
Drag to pan • wheel/pinch to zoom. Plots \(f(x)\), \(P_N(x)\), and optional intermediate partial sums.
Shows evenly spaced orders from \(0\) to \(N\) (plus \(N\)).
x: 0, y: 0, zoom(px/unit): 60
Series & steps
Enter \(f(x)\), \(a\), and order \(N\), then click Calculate.

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Frequently Asked Questions

What does the Taylor series about x=a represent?

It is a polynomial expansion that matches f(x) and its derivatives at x=a. The coefficients come from derivatives at a, using c_k = f^(k)(a)/k!.

How is a Maclaurin series different from a Taylor series?

A Maclaurin series is a Taylor series centered at a=0. Choosing a=0 makes the expansion powers of x instead of (x-a).

What does O((x-a)^(N+1)) mean in the result?

It indicates the size of the leftover terms after truncating at order N. The omitted remainder typically scales like (x-a)^(N+1) near the center.

How do I check the approximation accuracy at a specific point?

Enter an evaluation point x0 to see f(x0), P_N(x0), and the absolute error |f(x0) - P_N(x0)|. Increasing N usually improves accuracy near x=a.

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