Taylor Series

Subtopic: Series

Taylor series represent functions as infinite polynomials built from derivatives at a single point. They're the ultimate local approximation—capturing a function's behavior through its derivatives. Taylor series power numerical computation, physics approximations, and theoretical analysis.

Introduction

Polynomials are easy to compute—just addition and multiplication. What if we could write any function as a polynomial? Taylor series do exactly that (for well-behaved functions), expressing ƒ(x) as an infinite sum of powers of (x-a).

The coefficients come from derivatives: the more derivatives you include, the better the approximation near the expansion point.

Definition

The Taylor series of ƒ(x) centered at x=a:

ƒ(x)=(∑_n=0^∞)((ƒ(a)n)/n!)*(x−a)n

Expanded:

ƒ(x)=ƒ(a)+(ƒ^′)(a)*(x−a)+(ƒ^″)(a)/2!*(x−a)2+(ƒ^″)(a)/(3!)*(x−a)3+⋯

When a=0, it's called a Maclaurin series.

Key Taylor Series

ex=1+x+(x2)/(2!)+(x3)/(3!)+⋯=(∑_n=0^∞)((xn)/(n!))

sin(x)=x-(x3)/3!+(x5)/5!-…=(∑_n=0^∞)(((-1)n*x(2*n+1))/(2*n+1)!)

cos(x)=1-(x2)/2!+(x4)/4!-…=(∑_n=0^∞)(((-1)n*x(2*n))/(2*n)!)

ln(1+x)=x−(x2)/2+(x3)/3−⋯=(∑_n=1^∞)(((−1)(n+1)*xn)/n) (|x|<1)

1/(1−x)=1+x+x2+x3+⋯=(∑_n=0^∞)(xn) (|x|<1)

Worked Example

Find the Maclaurin series for ƒ(x)=ex.

All derivatives of ex equal ex. At x=0, we have ƒ(0)(n)=e0=1 for all n.

ex=(∑_n=0^∞)(1/n!)*xn=1+x+(x2)/2+(x3)/6+⋯

This converges for all x.

Taylor Polynomials

The nth-degree Taylor polynomial is the partial sum:

(T_n)(x)=(∑_k=0^n)((ƒ(a)(k))/k!)*(x−a)k

(T_1) is linear approximation (tangent line). (T_2) includes curvature. Higher degrees capture more local behavior.

Convergence

Not all Taylor series converge to their function. When they do, the radius of convergence R tells you how far from a the series works.

• ex, sin(x), cos(x): converge everywhere (R=∞)

• ln(1+x): converges for |x|<1 (R=1)

• 1/(1−x): converges for |x|<1 (R=1)

Error Estimate

The remainder (error) after n terms satisfies:

(R_n)(x)=(ƒ(c)(n+1))/(n+1)!*(x−a)(n+1)

for some c between a and x. This Lagrange form bounds the approximation error.

Applications

In physics, small-angle approximations such as sin(θ)≈θ come from Taylor series.

In numerical computing, functions like exp(x), sin(x), and cos(x) are computed using truncated Taylor series.

In calculus, Taylor series are used to evaluate limits and prove identities, such as Euler’s formula e(i*x)=cos(x)+i*sin(x).

In machine learning, second-order Taylor expansion underlies Newton's method for optimization.

Summary

A Taylor series expands ƒ(x) as (∑_n=0^∞)((ƒ(a)(n))/(n!))*(x−a)n. Key series include ex, sin(x), cos(x), ln(1+x), and 1/(1−x). Truncated Taylor polynomials approximate functions near the expansion point a. The radius of convergence determines where the series equals the function. Error bounds are given by the Lagrange remainder.