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Chapter 8: Q. 16 (page 669)

Is it possible for a power series to have (0,鈭�)as its interval converge? Explain your answer.

Short Answer

Expert verified

If there is a positive real integer 蚁, the series will therefore absolutely converge for every x鈭�(-蚁,蚁)

Step by step solution

01

Step 1. Given information.

Given, Is the interval convergence of a power series ever 0,鈭�

02

Explanation

It is impossible for a power series with an x value to have an interval of convergence of (0,鈭�)

The reason for this is if the power series 鈭�i=0鈭�akxkis the power series in x.

The series will therefore absolutely converge for every x鈭�(-蚁,蚁)if there is a positive real number 蚁.

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Most popular questions from this chapter

Show that the power series 鈭�k=0鈭�鈥�(鈭�1)k2k+1x2k+1converges conditionally when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

If f(x)=4x3-5x2+6x+1andP3(x) is the third Taylor polynomial for f at 鈭�1, what is the third remainder R3(x)? What is R4(x)? (Hint: You can answer this question without finding any derivatives.)

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

鈭�k=0鈭�(-1)k蟺2k+1(2k+1)!

Find the interval of convergence for power series:鈭�k=0鈭�-1k2k!x2k.

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?
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