Taylor series of single-variable and multi-variable functions

The Taylor series is a way to express a function as an infinite sum of terms, each of which is derived from the function’s derivative value at a specific point .

Consider the power series . We have , , ,  and so on. Therefore,

Eq32b is called a Taylor series, which approximates a function in the neighbourhood of the point . When , the Taylor series is also known as a Maclaurin series:

The input value of in eq32b represents points in the domain of that are near . In other words, we can express as , where is a constant, is a variable, and  represents a small change in that is scaled by . This alternate expression of is useful when dealing with multiple-variable functions.

For a multivariable function , the Taylor series about the point is



How do we use vector notation to express the function ?


If we let , then we write  in place of . In other words, we have , where . Consequently, can be regarded as

    1. a function of variables ,
    2. a function of a single point variable , or
    3. a function of a single vector variable .


To derive eq32c, let  and . Consider the function

which implies that .

is a multivariable function, whose input is the vector , which varies with . In the domain of , is the vector representing a point where is expanded, and is a fixed vector that determines the direction of the displacement from the point . In other words, is not a variable of , but a parameter that we choose before plotting . The function is therefore a single variable function of , and its points are of the form .

The Maclaurin series expansion of when is

According to the multivariable chain rule, , where  and . So,

The second derivative is

Using the multivariable chain rule again, we get

With reference to eq32d, with components . It follows that , and . Eq32g becomes

Furthermore, when , we have . So, eq32f and eq32h become



Substituting eq32i and eq32j in eq32e and noting that and , we have eq32c. Finally, the Maclaurin series of a multivariable function is



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