Taylor expansion/R/One variable/Introduction/Section

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Brook Taylor (1685-1731)

So far, we have only considered power series of the form . Now we allow that the variable may be replaced by a "shifted variable“ , in order to study the local behavior in the expansion point . Convergence means, in this case, that some exists, such that for

the series converges. In this situation, the function, presented by the power series, is again differentiable, and its derivative is given as in fact. For a convergent power series

the polynomials yield polynomial approximations for the function in the point . Moreover, the function is arbitrarily often differentiable in , and the higher derivatives in the point can be read of from the power series directly, namely

We consider now the question whether we can find, starting with a differentiable function of sufficiently high order, approximating polynomials (or a power series). This is the content of the Taylor expansion.

Let denote an interval,

an -times differentiable function, and . Then

is called the Taylor polynomial of degree for in the point .


is the constant approximation,

is the linear approximation,

is the quadratic approximation,

is the approximation of degree , etc. The Taylor polynomial of degree is the (uniquely determined) polynomial of degree with the property that its derivatives and the derivatives of at coincide up to order .

Let denote a real interval,

an -times differentiable function, and an inner point of the interval. Then for every point , there exists some such that

Here, may be chosen between

and .


This proof was not presented in the lecture.

The real sine function, together with several approximating Taylor polynomials (of odd degree).

Suppose that is a bounded closed interval,

is an -times continuously differentiable function, an inner point, and . Then, between and the -th Taylor polynomial, we have the estimate

The number exists due to fact, since the -th derivative is continuous on the compact interval . The statement follows, therefore, directly from fact.