Representing Functions with Taylor and Maclaurin Series We now discuss issues of convergence for Taylor series We begi


Representing Functions with Taylor and Maclaurin Series. We now discuss issues of convergence for Taylor series. We begin by showing how to find a Taylor series .



Common Functions Expressed as Taylor Series At this point
we have derived Maclaurin series for exponential

trigonometric.

and logarithmic functions
as


1 Maclaurin series are named after the Scottish mathematician Colin Maclaurin The Maclaurin series of a function up to




A series expansion is a representation of a particular function as a sum of powers in one of its variables
or by a sum of powers of another usually elementary


I recently came across the following












.

1 The factorials in the denominators reminded me of a Taylor Series In particular
I found that .



Representing Functions with Taylor and Maclaurin Series. We now discuss issues of convergence for Taylor series. We begin by showing how to find a Taylor .







How to obtain the Maclaurin Series of a Function In general
a well behaved function.



f x and all its derivatives are finite at x


0 will be expressed as an infinite .







Explore the Maclaurin series.





a special version of the Taylor series used to help approximate complicated mathematical functions. Review the Taylor series.



.



Maclaurin Series. The Maclaurin series is a special case of the Taylor series for a continuous function at x.



0. It is a summation of all the derivatives of a .



Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series. where is a gamma function.

is a Bernoulli number is an Euler number and is a Legendre polynomial


Our first goal in this section is to determine the Maclaurin series for the function f x

1

x r for all real numbers r The Maclaurin series for this function is known as the binomial series We begin by conside


Explore the Maclaurin series

a special version of the Taylor series used to help approximate complicated mathematical functions. Review the Taylor series. discover more about the Maclaurin series



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A series expansion is a representation of a particular function as a sum of powers in one of its variables.



or by a sum of powers of another usually elementary .





Find the Maclaurin series for \ f x \cos x\. Use the ratio test to show that the interval of convergence is \ −∞.

∞ \ Show that the Maclaurin series converges to \ \cos x\ for all real numbers \ x\ Hint Use the Maclaurin polynomi
2n.



\


13 Series We have seen that some functions can be represented as series

which may give valuable information about the function So far
we have seen only those examples that result from manipulation of our one fundamental example.



the geometric series. We would like to start with a given function and produce a series to .





In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges.



is increasing or decreasing.



or if the sequence is bounded We will then define just what an infinite series is and discuss many of the basic concep

including many



Differential equations are made easy with Taylor series Taylor’s series is an essential theoretical tool in computati