Module 24  Power Series 
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest 
Lesson 24.3: Taylor Series 
In Lesson 24.2 you found Maclaurin series that approximate functions near x = 0. This lesson investigates how to find a series that approximates a function near x = a, where a is any real number. A Taylor Series Given a function f that has all its higher order derivatives, the series , where is called the Taylor series for f centered at a. The Taylor series is a power series that approximates the function f near x = a. The partial sum is called the nthorder Taylor polynomial for f centered at a. Every Maclaurin series, including those studied in Lesson 24.2, is a Taylor series centered at zero. The Taylor Polynomial of e^{x} Centered at 1 The secondorder Taylor polynomial centered at 1 for the function f(x) = e^{x} can be found by using a procedure similar to the procedure given in Lesson 24.2. The coefficient of the term (x  1)^{k} in the Taylor polynomial is given by . This formula is very similar to the formula for finding the coefficient of x^{k} in a Maclaurin polynomial where the derivative is evaluated at 0. In this Taylor polynomial, the derivative is evaluated at 1, the center of the series. The coefficients of the secondorder Taylor polynomial centered at 1 for e^{x} are f(1) = e f '(1) = e
So the secondorder Taylor polynomial for e^{x} centered at 1 is , and near x = 1, e^{x} P_{2}(x). The Taylor series for e^{x} centered at 1 is similar to the Maclaurin series for e^{x} found in Lesson 24.2. However, the terms in the Taylor series have powers of (x  1) rather than powers of x and the coefficients contain the values of the derivatives evaluated at x = 1 rather than evaluated at x = 0.
Graphing the function and the polynomial illustrate that the polynomial is a good approximation near
The secondorder Maclaurin polynomial you found in Lesson 24.2, , is tangent to f(x) = e^{x} at x = 0 and has the same concavity as f(x) = e^{x} at that point. The polynomial , which is centered at x = 1, is tangent to f(x) = e^{x} at x = 1 and has the same concavity as f(x) = e^{x} at that point. 24.3.1 Find the secondorder Taylor polynomial centered at 1 for the function f(x) = ln x. Graph this polynomial together with f(x) = ln x. Click here for the answer. Other Taylor Series The Taylor series for one function can be used to find a Taylor series for a related function. The thirdorder Taylor polynomial centered at 1 for f(x) = ln x is . The derivative of f(x) = ln x is . The derivative of p(x) gives the secondorder Taylor polynomial for centered at 1. 24.3.2 Find the secondorder Taylor polynomial for centered at 1 using the derivative of p(x) and graph it with . Click here for the answer. The Taylor Series for cos x^{2} Other modifications of a Taylor series yield other Taylorseries. For example, replacing each x with x^{2} in the Taylor series for f(x) = cos(x) gives the Taylor series for g(x) = f(x^{2}) = cos(x^{2}).

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