   Chapter 11.8, Problem 40E

Chapter
Section
Textbook Problem

Suppose that the power series ∑ c n ( x − a ) n satisfies c n ≠ 0 for all n. Show that if lim n → ∞ | c n / c n + 1 | exists, then it is equal to the radius of convergence of the power series.

To determine

To show:

The

limncncn+1

exists and it is equal to the radius of convergence of the power series .

Explanation

1) Concept:

i) For a power series n=0cnx-an, there is a positive number R such that the series converges if x-a<R and diverges if x-a>R, this number R is called the radius of convergence and the series is centered at  x=a.

ii) The ratio test: If

limnan+1an=L<1,

then the series an is absolutely convergent, and if

limnan+1an=L>1,

the series diverges, the radius of convergence by ratio test is denoted as R=1/L

2) Given:

The power series n=0cnx-an satisfies cn0 for all  n

3) Calculation:

Given that the power series n=0cnxn satisfies cn0 for all  n.

So nth terms is, an= cnx-an

Therefore, by using the ratio test,

L=limnan+1an

=limncn+1x-an+1cnx-an

=limnx-an+1x-ancncn+1

=limn(x-a)cn/cn+1

L =x-alimncncn+1

As cn0  so cn+10

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