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Chapter 8: Q. 8 (page 669)

Show that 鈭�k=0鈭�鈥�11+2kxk, the power series in xfrom Example 1, diverges when x=-2

Short Answer

Expert verified

Ans: The power series鈭�k=0鈭�鈥�11+2kxkdiverges whenx=-2.

Step by step solution

01

Step 1. Given information.

given,

鈭�k=0鈭�鈥�11+2kxk

02

Step 2. We evaluate the series when x=-2

So,

鈭�k=0鈭�鈥�11+2k(鈭�2)k=鈭�k=0鈭�鈥�(鈭�1)k2k1+2k

Therefore, when k鈫�鈭�,limx鈫�鈭�鈥�(鈭�1)k2k1+2k=1

Thus by the divergence test, the power series 鈭�k=0鈭�鈥�11+2kxkdiverges whenx=-2

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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)

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What is x0if (p,q)is the interval of convergence for the power series 鈭�k=0鈭�鈥�akx鈭�x0k?

Exercise 64-68 concern with the bessel function.

What is the interval for convergence forJ0(x)?

Prove that if the power series 鈭�k=0鈭�鈥�akxkhas a positive and finite radius of convergence p, then the series 鈭�k=0鈭�鈥�akx2khas a radius of convergence p.

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.)
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