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Chapter 8: Q. 17 (page 669)

Let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a positive and finite radius of convergence p. Explain why the ratio test for absolute convergence will fail to determine the convergence of this power series when x=por when x=-p.

Short Answer

Expert verified

Ans: The ratio test for absolute convergence fails at the endpoints of the interval of convergence.

Step by step solution

01

Step 1. Given information.

given,

鈭�k=0鈭�鈥�akxk

02

Step 2. Solution.

If 鈭�k=0鈭�鈥�akxkis a power series in xwith a positive and finite radius of convergence pthen the limit localid="1649392252268" limk鈫�鈭�鈥�bk+1bk=limk鈫�鈭�鈥�ak+1ak蚁which is equal to 1.

03

Step 3. Thus,

The ratio test for absolute convergence fails at the endpoints of the interval of convergence.

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