Taylor series definition and related expansion

Taylor series, Euler's formula, trigonometric function

Taylor series definition:

If the function f (x) has an order derivative until (n + 1) in a certain domain of a vertex, the n-order Taylor Formula of f (x) in this neighborhood is:

Among them, it is called the remainder of the Langran.

The preceding function expansion is called the Taylor series.

The role of the Taylor series in the expansion of the Power Series:

In Taylor's formula, take the following:

This level is called the mclulin level. The mclulin series of the f (x) function is the power series of X, so this expansion is unique and must be consistent with the mclulin series of f (x.

Note: If the mclulin series of f (x) converges in a certain domain of a vertex, it does not necessarily converge in F (x ). Therefore, if f (x) has a derivative of each order, then although the R/W series of f (x) can be made, can this series converge within a certain region, and whether to converge to f (x) requires further verification.

Several important Taylor series. ParametersXThey are still valid when they are plural.

• Exponential Function and natural logarithm:

• Geometric series:

• Binary theorem:

• Trigonometric function:

• Hyperbolic function:

• Longman functions:

 

C (α,N) Is the binary coefficient.

Tan (X) And Tanh (X) In expansionBKIt is the bernuoli number.

SEC (X) In expansionEKIt is the number of OLA.

Point on a unit circle on the complex plane. When the angle between the point and the real axis is θ, this point can be expressed

E is the basis of the natural logarithm. This formula is called the Euler's formula. E can be defined

Relationship Between Euler's formula and trigonometric function

Expanded by Taylor series

 

From: http://blog.163.com/xiangyuan_122/blog/static/280073612009112810632928/