Taylor Series Formula
The Formula
When to use: Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.
Quick Example
e^1 \approx 1 + 1 + 0.5 + 0.167 + 0.042 + \cdots = 2.718\ldots
Each term improves the approximation.
Notation
What This Formula Means
When a = 0, it's called a Maclaurin series.
Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.
Formal View
Worked Examples
Example 1
easySolution
- 1 The Taylor series formula is f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n. Compute the derivatives of e^x at x=0.
- 2 Since every derivative of e^x is e^x, we have f^{(n)}(0) = e^0 = 1 for all n.
- 3 Substitute into the formula: e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots
Answer
Example 2
hardCommon Mistakes
- Forgetting the n! in the denominator: the coefficient of (x-a)^n is \frac{f^{(n)}(a)}{n!}, not f^{(n)}(a).
- Confusing Taylor polynomials (finite, approximate) with Taylor series (infinite, exact within radius of convergence): 1 + x + x^2 approximates \frac{1}{1-x}, but equals it only as 1 + x + x^2 + x^3 + \cdots.
- Assuming the Taylor series always converges to the function: there exist functions (like f(x) = e^{-1/x^2} for x \neq 0, f(0) = 0) whose Taylor series converges everywhere but equals the function only at the center point.
Why This Formula Matters
Taylor series are how calculators compute \sin, \cos, e^x, and \ln. They are used in physics for linearization and perturbation theory, in numerical methods for approximation algorithms, and in analysis for proving theorems about functions.
Frequently Asked Questions
What is the Taylor Series formula?
When a = 0, it's called a Maclaurin series.
How do you use the Taylor Series formula?
Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.
What do the symbols mean in the Taylor Series formula?
T_n(x) = Taylor polynomial of degree n. R_n(x) = remainder (error) term.
Why is the Taylor Series formula important in Math?
Taylor series are how calculators compute \sin, \cos, e^x, and \ln. They are used in physics for linearization and perturbation theory, in numerical methods for approximation algorithms, and in analysis for proving theorems about functions.
What do students get wrong about Taylor Series?
A Taylor series may only equal the function within a certain radius of convergence. Outside this radius, the series diverges and is useless. For example, \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots only converges for -1 < x \leq 1.
What should I learn before the Taylor Series formula?
Before studying the Taylor Series formula, you should understand: derivative, differentiation rules, infinite geometric series, convergence divergence.