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Chapter 8: Q. 5 (page 700)

Let f(x)=鈭�k=0鈭�akx-x0kand let Gbe the antiderivative for fwith the property that Gx0=7. Find the Taylor series inx0 for G.

Short Answer

Expert verified

The Taylor series in x0for G is G(x)=鈭�k=0鈭�akk+1x-x0k+7.

Step by step solution

01

Step 1. Given information

The function is f(x)=鈭�k=0鈭�akx-x0k.

Find the Taylor series inx0forG?

02

Step 2. Simplification

Let's take a look at the function's power seriesf(x)=鈭�k=0鈭�akx-x0k

Given that Gis the antiderivative of f.

G(x)=鈭�f(x)dx

Where Gx0=7

Thus, the Taylor series in x0forGis

G=鈭�鈭�k=0鈭�akx-x0kdx=鈭�k=0鈭�ak鈭�x-x0kdx=鈭�k=0鈭�akx-x0k+1k+1+C

Where C is the constant of integration,

03

Step 3. Find the Taylor series

Change the index of the power series you've created now.

So,

G=鈭�k=0鈭�akk+1x-x0k+C

The value C can be found in this case by usingGx0=7

Thus,

Gx0=鈭�k=0鈭�akk+1x0-x0k+C

It signifies that

7=C

Therefore,G(x)=鈭�k=0鈭�akk+1x-x0k+7

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Most popular questions from this chapter

Show that the power series 鈭�k=1鈭�鈥�(鈭�1)kk2xkconverges absolutely when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

Given a function f and a Taylor polynomial for fat x0, what is meant by the nth remainder Rn(x)? What does Rn(x)measure?

Let f be a twice-differentiable function at a point x0. Using the words value, slope, and concavity, explain why the second Taylor polynomial P2(x)might be a good approximation for f close to x0.

What is Lagrange鈥檚 form for the remainder? Why is Lagrange鈥檚 form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?

Use an appropriate Maclaurin series to find the values of the series in Exercises 17鈥�22.
鈭�k=0鈭�-蟺kk!
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