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Taylor approximation and error

How well a polynomial stands in for a function, and how far you can trust it.

Near the centre the Taylor polynomial is nearly the function; the error is controlled by the next derivative and the distance from the centre. This is why calculators can compute sin: a handful of terms inside a small interval, plus symmetry.

Apstrādāts piemērs: taylor series of sin(x)

Calculus · Taylor-Series

Taylor series of sin(x)

\sin{\left(x \right)}

Solis pēc kārtas

  1. f(x) = \sin{\left(x \right)}

    Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

  2. f^{(0)}(0) = \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 0 at the centre.

  3. f^{(1)}(0) = \cos{\left(x \right)}\big|_{x=0} = 1

    Derivative 1 at the centre.

  4. f^{(2)}(0) = - \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 2 at the centre.

  5. f^{(3)}(0) = - \cos{\left(x \right)}\big|_{x=0} = -1

    Derivative 3 at the centre.

  6. f^{(4)}(0) = \sin{\left(x \right)}\big|_{x=0} = 0

    Derivative 4 at the centre.

  7. f^{(5)}(0) = \cos{\left(x \right)}\big|_{x=0} = 1

    Derivative 5 at the centre.

  8. \frac{x^{5}}{120} - \frac{x^{3}}{6} + x

    Assemble the terms up to degree 5.

Atbildēt uz šo jautājumu
\frac{x^{5}}{120} - \frac{x^{3}}{6} + x + O((x-0)^{6})

Symbols used here

\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varepsilon,\ \delta
epsilon, delta
Small positive tolerances in the definition of a limit.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.

How to: Taylor approximation and error

  1. Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.
  2. Derivative 0 at the centre.
  3. Derivative 1 at the centre.
  4. Derivative 2 at the centre.
  5. Derivative 3 at the centre.
  6. Derivative 4 at the centre.
  7. Derivative 5 at the centre.
  8. Assemble the terms up to degree 5.

Questions people ask

What is the ε–δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on — infinitely many halves.

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Vairāk Real Analysis

Sequences and their limitsConvergence of seriesImproper integralsFunctions of a complex variableContinuity and differentiability, rigorouslySequences of functions and uniform convergence