Q: How large is the Taylor series remainder (error)?

The remainder after N terms is given by the Lagrange form Rₙ(x) = f^(N+1)(ξ)/(N+1)! · (x−a)^(N+1) for some ξ between a and x. The error grows as |x−a| increases and shrinks as N increases. For sin x with N=5 at a=0, the error is extremely small for |x| π.

Question 1

What is the radius of convergence of a Taylor series?

Accepted Answer

The radius of convergence R is the radius centered at the expansion point a within which |x−a| < R guarantees convergence of the Taylor series. It is determined by the Cauchy-Hadamard formula 1/R = lim|cₙ|^(1/n). For example: eˣ has R=∞ (converges everywhere), ln(1+x) at a=0 has R=1 (converges only for |x|<1), and 1/(1-x) also has R=1. Outside the radius, the series diverges.

Question 2

How large is the Taylor series remainder (error)?

Accepted Answer

The remainder after N terms is given by the Lagrange form Rₙ(x) = f^(N+1)(ξ)/(N+1)! · (x−a)^(N+1) for some ξ between a and x. The error grows as |x−a| increases and shrinks as N increases. For sin x with N=5 at a=0, the error is extremely small for |x| < π/4 but becomes significant near |x| > π.

Question 3

What is the difference between Taylor and Maclaurin series?

Accepted Answer

A Maclaurin series is a special case of a Taylor series where the expansion point a=0. It gives f(x) = f(0) + f'(0)x + f''(0)x²/2! + ... In this tool you can set a anywhere from −3 to +3; setting a=0 gives the Maclaurin series. Changing a shifts the region of convergence (within the radius) to be centered at a.

Question 4

How is Taylor series used in CAE and numerical computation?

Accepted Answer

Taylor series underpins many CAE methods: (1) FEM shape functions approximate displacement fields locally as polynomials. (2) Newton's method linearizes nonlinear equations using the first-order Taylor expansion — the tangent stiffness matrix is exactly this. (3) Finite difference schemes derive their accuracy from Taylor expansions of the truncation error. (4) Small-angle approximation sin θ ≈ θ (1st-order Taylor) linearizes rotational kinematics. (5) Creep/relaxation functions use exponential expansions.

Taylor Series Approximation Visualizer

Taylor Series Formula

CAE Engineering Applications