Module 24  Power Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 24.2: Maclaurin Series  
In Lesson 24.1 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 best converges to that function for values of x near zero. Approximating a Function Near x = 0
Suppose f is some function. A seconddegree polynomial p(x) = ax^{2} + bx + c that satisfies p(0) = f(0), Let f(x) = e^{x} and p(x) = ax^{2} + bx + c. Use the fact that f and p are equal at x = 0 to find the value of c.
If c = 1, the function and the polynomial have the same value at x = 0, so p(x) = ax^{2} + bx + 1. Using a similar procedure, set the first derivatives equal when x = 0 and solve for b.
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 when x = 0 and solve for a.
If a = 1/2, the function and the polynomial will have the same concavity at x = 0. 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} when x = 0, and the quadratic polynomial is a good approximation for y = e^{x} when x is near 0. 24.2.1 Create a table of values for f(x) = e^{x} and p(x) = (1/2)x^{2} + x + 1 with TblStart = 1 and Tbl = 0.1. Compare the values of e^{x} and the quadratic polynomial when x = 0.5 and when x = 0.5. Click here for the answer. The General Procedure for Finding Maclaurin Series The procedure for finding the coefficients of a power series that approximates a function near x = 0 is summarized below, where the function y = f(x) is to be approximated by the power series polynomial .
The series found in this fashion is called the Maclaurin series for the function f. Maclaurin series are often useful for integrating functions that do not have elementary antiderivates, for solving differential equations, and for approximating functions by polynomials. Coefficients in a Maclaurin Series Letting f^{(k)}(x) denote the kth derivative of f, it can be shown that in a Maclaurin series the coefficient of the term that contains the kth power of x is given by The table below shows the first four coefficients of the Maclaurin series for f(x) = e^{x}.
Therefore, the Maclaurin series for e^{x} is . The Maclaurin series above is more than an approximation of e^{x}, it is actually equal to e^{x} on the interval of convergence, . Every Maclaurin series is centered at x = 0, the interval of convergence is centered at x = 0, and the Maclaurin series values and the function values are very close near x = 0. The partial sum is called the nthorder Maclaurin polynomial for f, and if n is relatively large, an nthorder Maclaurin polynomial often approximates the function f well on the interval of convergence. 24.2.2 Find the fifthorder Maclaurin polynomial for y = ln(1 + x) and interpret the result. Click here for the answer. 24.2.3 Graph y = ln(1 + x) and the fifthorder polynomial from 24.2.2. Click here for the answer. 

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