# Power series, Taylor series and Functions

Since recently I am dealing with numerical solution of differential equation, I recalled that not long time ago I’ve actually learned about and Euler’s equation. They should be close related to the differential equation topic. So let’s review some of the conclusions I learned from the book: “オイラーの公式がわかる”.

(A series doesn’t need to have a variable in it: a infinite series of numbers can be a series.)

### Power series

In mathematics, a power series (in one variable) is an infinite series of the form:

where represents the coefficient of the nth term and c is a constant. In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form:

### Use a power series to generate a function

Suppose we have an arbitrary function . We can use a power series to approximate the function in the following way:

The power series takes the form:

Since should equal to

Thus:

In a similar fashion, we differentiate the function once and should equal to

So:

Repeat this algorithm one more time and we will find:

Finally the general form gives:

### Taylor series

In fact, we just derived the Taylor series **near 0** for .

where

Use such a technique, we can find the Taylor series of and
(**again, near 0**):

(The fact that is odd and is even could also be found out from right hand side!)

今日はここまでお疲れ様でした！