Module 22  Power Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 22.2: Maclaurin Series  
In Lesson 22.1 you explored several power series and their relationships to the functions to which they converge. In this lesson you will start with a function and find the power series that converges to that function for values of x near zero. This type of series is called a Maclaurin series. Finding a Quadratic Polynomial that Approximates a Function Near x = 0
It turns out that finding a seconddegree polynomial p(x) = ax^{2} + bx + c that satisfies p(0) = f(0), The procedure below illustrates the method by finding a quadratic polynomial that satisfies these conditions for the function f(x) = e^{x}.
Find the value of c when the two function values are equal at x = 0.
If c = 1, the function and the polynomial have the same value at x = 0, so p(x) = ax^{2} + bx + 1 Set the first derivatives equal and solve for b when x = 0.
If b = 1, the function and the polynomial will have the same slope at x = 0, so p(x) = ax^{2} + x + 1. Set the second derivatives equal and solve for a when x = 0.
If a = 1/2, the function and the polynomial will have the same concavity at x = 0. Therefore, Displaying Graphs of the Function and the Approximating Quadratic Polynomial
The parabola has the same value, the same slope and the same concavity as y = e^{x} at x = 0. The quadratic polynomial is a good approximation for y = e^{x} at x = 0. 22.2.1 Create a table of values for y = e^{x} and with tblStart = 1 and tbl = 0.1. Compare the values of y = e^{x} and the quadratic polynomial when x = 0.5 and when x = 0.5. Click here for the answer. Outlining the General Procedure of Finding Maclaurin Series The general idea for finding the coefficients of a power series
that approximates a function y = f(x) near x = 0 is first to set the value of the power series equal to the value of f(x) at x = 0 and solve for c_{0}. Then, set all of the derivatives of the power series (beginning with the first) equal to the corresponding derivative of f(x) at x = 0 and solve for the remaining coefficients of the power series. The series found is this fashion is called the Maclaurin series for f(x). Letting f ^{(k)}(x) represent the kth derivative of f, the coefficient of the term that contains the kth power of x in a Maclaurin series is
This method produces the following power series for y = e^{x} :
The Maclaurin series above is more than an approximation of e^{x}, it is equal to e^{x} on the interval of convergence (– , ). Every Maclaurin series is centered at 0 and the interval of convergence is centered at 0. When the Maclaurin series approximates a function, the series values and the function values are very close near x = 0. The partial sum
is the nthorder Maclaurin polynomial for f. Generating Maclaurin Polynomials on a TI89 The TI89 can generate Maclaurin polynomials for a function f using the taylor( command.
Generate the secondorder Maclaurin polynomial for e^{x} using the taylor( command.
22.2.2 Use the taylor command to find the fifthorder Maclaurin polynomial for 22.2.3 Graph y = ln(1 + x) and the fifthorder polynomial from 22.2.2 together. Click here for the answer. 

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