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Chapter 8: Q. 42 (page 692)

In Exercises 41-48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ . In Exercises 37-44 give Lagrange's form for the remainder $R_{4} (x)$ .

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Most popular questions from this chapter

What is a power series in $x - x_{0}$ ?

Find the interval of convergence for each power series in Exercises 21–48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

$\sum_{k = 0}^{\infty} {(- 1)}^{k} \frac{k^{2}}{3^{k}} {(x - \frac{1}{2})}^{k}$

Let $\sum_{k = 0}^{\infty} a_{k} x^{k}$ be a power series in x with an interval of convergence $[- 2, 2)$ . What is the radius of convergence of the power series $\sum_{k = 0}^{\infty} a_{k} (x - 3)^{k}$ ? Justify your answer.

The second-order differential equation

$x^{2} y^{''} + x y^{'} + (x^{2} - p^{2}) = 0$

where p is a nonnegative integer, arises in many applications in physics and engineering, including one model for the vibration of a beaten drum. The solution of the differential equation is called the Bessel function of order p, denoted by $J_{p} (x)$ . It may be shown that $J_{p} (x)$ is given by the following power series in x:

$J_{p} (x) = \sum_{k = 0}^{\infty} \frac{(- 1)^{k}}{k! (k + p)! 2^{2 k + p}} x^{2 k + p}$

Find and graph the first four terms in the sequence of partial sums of $J_{o} (x)$ .

In Exercises 41–48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ .

$48. \sin 3 x, \frac{π}{6}$

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In Exercises 41-48 in Section 8.2, you were asked to find the fourth Taylor polynomial P4(x)for the specified function and the given value of x0. In Exercises 37-44 give Lagrange's form for the remainder R4(x).

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In Exercises 41-48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ . In Exercises 37-44 give Lagrange's form for the remainder $R_{4} (x)$ .