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Chapter 8: Q. 23 (page 692)

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$
$\cos x$

Short Answer

Expert verified

The required answer is $R_{4} (x) = - \frac{\sin c}{120} x^{5}$

Step by step solution

01

Step 1. Given Information

The given function is $f (x) = \cos x$

02

Step 2. Explanation

Using the Lagrange form of remainder, we have,

$R_{n} (x) = \frac{f^{n + 1} (c)}{(n + 1)!} x^{n + 1} R_{4} (x) = \frac{f^{5} (c)}{5!} x^{5}$

Now, we will find fifth derivative of the function.

$f^{1} (x) = - \sin x f^{2} (x) = - \cos x f^{3} (x) = \sin x f^{4} (x) = \cos x f^{5} (x) = - \sin x$

Thus, we get,

$R_{4} (x) = - \frac{\sin c}{120} x^{5}$

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Most popular questions from this chapter

Prove that if $(- p, p]$ is the interval of convergence for the series $\sum_{k = 0}^{\infty} a_{k} x^{k}$ , then the series converges conditionally at $p$ .

Find the interval of convergence for power series: $\sum_{k = 1}^{\infty} \frac{1}{k} {(x + 2)}^{k}$

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_{4} (x)$

$\sin x$

Let $\sum_{k = 0}^{\infty} a_{k} {(x - x_{0})}^{k}$ be a power series in $x - x_{0}$ with a finite radius of convergence $p$ . Prove that if the series converges absolutely at $x_{0} \pm p$ , then the series converges absolutely at the other value as well.

The second-order differential equation

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

where p is a non-negative 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:

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

What is the interval of convergence for $J_{p} (x)$ where p is a non-negative integer

See all solutions

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