Taylor series single variable

A set of nonlinear equations can be solved by combining a Taylor series linearization with the linear equation-solving approach discussed above. For solving a single nonlinear equation, h(x) = 0, Newton s method applied to a function of a single variable is the well-known iterative procedure... 

A function is called analytic at a point if it is possible to expand it in a Taylor series valid in some neighbourhood of tbe point. This is equivalent to saying that the ( unction is single-valued and possesses derivatives of all orders at the point in question. In the equations we shall consider the coefficients will be analytical functions of the independent variable except possibly at certain isolated points. 

As a reminder, we write the Taylor series expansion for a function J x) of one single variable x. 

Refer to equations (L.5). For a single equation (and variable),/(jc) = 0, Newton s method uses the expansion of / ( c) in a first-order Taylor series about a reference point (a starting guess for the solution) jco. 

---