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Chapter 8: Q. 45 (page 692)

In Exercises 49-54in Section 8.2you were asked to find the Taylor series for the specified function at the given value of x0. In Exercises 45-50find the Lagrange's form for the remainder Rn(x), and show that limn鈫�鈭�Rn(x)=0on the specified interval.

cosx,蟺2,鈩�

Short Answer

Expert verified

That, which isRn(x)=x-蟺2n+1(n+1)!proved.

Step by step solution

01

Given Information

Let us consider the functionf(x)=cosx

02

Lagrange's Form

If f(x)=cosx, we know that for every n鈮�0,f(n+1)(x)is one of the four function 卤sinxand 卤cosx. So, for any of the four functions, f(n+1)(c)鈮�1, for every value of x, so using the Lagrange's form for the remainder, we have

Rn(x)=f(n+1)(c)(n+1)!xn+1

03

Proof

Since the Taylor series for the function f(x)=cosxat x=蟺2is

Pn(x)=鈭�k=0鈭�(-1)n+1(2n+1)!x-蟺22n+1

Therefore,

Rn(x)=x-蟺2n+1(n+1)!

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Most popular questions from this chapter

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

ln(1+x)

Complete Example 4 by showing that the power series 鈭�k=0鈭�鈥�(鈭�1)kk3k+1(2x鈭�3)kdiverges when x=2.

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

鈭�k=1鈭�2k+1kxk

Use an appropriate Maclaurin series to find the values of the series in Exercises 17鈥�22.

鈭�k=0鈭�(-1)k2k+1132k+1

In Exercises 41鈥�48 find the fourth Taylor polynomial P4(x)for the specified function and the given value of x0.

48.sin3x,蟺6
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