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Chapter 8: Q. 11 (page 692)

What is Taylor鈥檚 Theorem?

Short Answer

Expert verified

If f be a function that can be differentiated (n+1) times in some open intervalI that contains the pointx0andalsoRn(x)represents the nth remainder for the function atx=x0.

Step by step solution

01

Step 1. Given Information

The given term is Taylor's polynomial

02

Step 2. Explanation

Taylor's theorem state that if f be a function that can be differentiated (n+1) times in some open intervalI that contains the point x0andalsoRn(x) represents the nth remainder for the function at x=x0.

Then the nth remainder for function is given by,

Rn(x)=鈭�x0xx-tnn!fn+1(t)dt

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Most popular questions from this chapter

If f(x) is an nth-degree polynomial and Pn(x) is the nth Taylor polynomial for fat x0, what is the nth remainder Rn(x)? What is Rn+1(x)?

The second-order differential equation

x2y''+xy'+x2-p2=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 Jp(x). It may be shown that Jp(x)is given by the following power series in x :

Jp(x)=鈭�k=0鈭�(-1)kk!(k+p)!22k+px2k+p

What is the interval of convergence for J1(x)?

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48.

54.x3,1

Find the interval of convergence for power series:鈭�k=0鈭�kk!4x+7k

In Exercises 23鈥�32 we ask you to give Lagrange鈥檚 form for the corresponding remainder, R4(x)

cosx
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