Taylor series

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Definition

About a general point

Suppose f is a function that is infinitely differentiable at a point x_0 in its domain. The Taylor series of f about x_0 is the power series given as follows:

\sum_{k=0}^\infty \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k

Here's a version with the first few terms written explicitly:

f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2}(x - x_0)^2 + \frac{f'''(x_0)}{6}(x - x_0)^3 + \dots

About the point 0

In the special case of the above definition where x_0 = 0 (and in particular f is infinitely differentiable at 0), the Taylor series is as follows:

\sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!} x^k

Here's a version with the first few terms written explicitly:

f(0) + f'(0)x + \frac{f''(0)}{2}x^2 + \frac{f'''(0)}{6}x^3 + \dots +

Well defined on germs of a functions

The Taylor series operator about a point x_0 can be thought of as a mapping:

(Germs of C^\infty-functions defined about x_0) \to (Formal power series centered at x_0)

In fact, this mapping is a \R-algebra homomorphism that commutes with the differential structure.

Here, two functions f and g are said to have the same germ about a point x_0 if there is an open interval U containing x_0 such that f(x) = g(x) \ \forall x \in U.

Relation with Taylor polynomials

The Taylor series can be viewed as a limit of Taylor polynomials. The n^{th} Taylor polynomial for a function f at a point x_0 in the domain is the truncation of the Taylor series to powers up to the n^{th} power. If we denote the polynomial by P_n(f;x_0), it is given as:

P_n(f;x_0) = x \mapsto \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!}(x - x_0)^k

Note that this is a polynomial of degree at most n. The degree is exactly n if and only if f^{(n)}(x_0) \ne 0.

Whether or not the Taylor series of a function converges to the function is determined by whether or not the sequence of Taylor polynomials of the function converges to the function.

Computation of Taylor series

To calculate the Taylor series of a function f at a point x_0, we use the following procedure:

  • Compute formal expressions for f,f',f'',\dots, i.e., f and all its derivatives, at a generic point.
  • Evaluate all these at x_0.
  • Plug the result into the Taylor series formula.

Example of exponential function

For further information, refer: Exponential function#Taylor series

Consider the exponential function:

f = \exp

i.e., f(x) = e^x.

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

  • Formal expressions for f,f',f'',\dots: These are \exp,\exp,\exp,\dots. The sequence is a constant sequence of functions with all its members equal to \exp.
  • Evaluate at 0: Since \exp(0) = 1, we get 1,1,1,\dots. The sequence is a constant sequence with value 1 in all places.
  • Plug the result into the Taylor series formula: We get:

Taylor series for \exp(x) is \sum_{k=0}^\infty \frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \dots

Note that it is also true that the Taylor series for the exponential function converges to the exponential function everywhere; this is because the function is globally analytic. However, this fact is not a priori obvious, and we are not asserting it as part of the computation of the Taylor series.

Example of cosine function

For further information, refer: Cosine function#Taylor series

Consider the cosine function:

f = \cos

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

  • Formal expressions for f,f',f'',\dots: These are \cos, -\sin, -\cos, \sin, \cos, -\sin, -\cos, \sin, \dots. The sequence is periodic with period 4.
  • Evaluate at 0: Since \cos 0 = 1, \sin 0 = 0, the values are 1,0,-1,0,1,0,-1,0,\dots. The sequence is periodic with period 4.
  • Plug the result into the Taylor series formula: We get:

Taylor series for \cos x is 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots which can be rewritten compactly as \sum_{k=0}^\infty \frac{(-1)^kx^{2k}}{(2k)!}. Note that the k here is half the exponent on x, so this is a little different from the usual way of writing Taylor series.

Note that it is also true that the Taylor series for the cosine function converges to the cosine function everywhere; this is because the function is globally analytic. However, this fact is not a priori obvious, and we are not asserting it as part of the computation of the Taylor series.

Example of sine function

For further information, refer: Sine function#Computation of Taylor series

Consider the sine function:

f = \sin

We want to compute the Taylor series of this function at 0.

Applying the procedure above, we get:

  • Formal expressions for f,f',f'',\dots: These are \sin, \cos, -\sin, -\cos, \sin, \cos, -\sin, -\cos, \dots. The sequence is periodic with period 4.
  • Evaluate at 0: Since \sin 0 = 0, \cos 0 = 1, the values are 0,1,0,-1,0,1,0,-1,\dots. The sequence is periodic with period 4.
  • Plug the result into the Taylor series formula: We get:

Taylor series for \sin x is x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots which can be rewritten compactly as \sum_{k=0}^\infty \frac{(-1)^kx^{2k + 1}}{(2k + 1)!}. Note that the k here is roughly half the exponent on x, so this is a little different from the usual way of writing Taylor series.

Note that it is also true that the Taylor series for the sine function converges to the sine function everywhere; this is because the function is globally analytic. However, this fact is not a priori obvious, and we are not asserting it as part of the computation of the Taylor series.

Facts

Preservation of structure

Together, the first three facts show that the Taylor series operator is a homomorphism of \R-algebras that commutes with the differential structure. The fourth fact show that it preserves an additional structure:

Convergence to the original function

A function whose Taylor series at a point converges to the function in an open interval centered at the point is termed a locally analytic function at the point. If the Taylor series converges to the function everywhere, the function is termed a globally analytic function. We have that locally analytic not implies globally analytic.