   Chapter 11.8, Problem 39E

Chapter
Section
Textbook Problem

# Show that if lim n → ∞ | c n | n = c , where c ≠ 0, then the radius of convergence of the power series ∑ c n x n is R = 1/c.

To determine

To show: The radius of convergence of the power series cnxn is R=1c, if limn|cn|n=c.

Explanation

Formula Used:

Root test:

(1). If limn|an|n=L<1, then the series n=1an is absolutely convergent and therefore convergent.

(2). If limn|an|n=L>1 or limn|an|n=, then the series n=1an is divergent.

(3). If limn|an|n=1, the root test is inconclusive.

Calculation:

Given that limn|cn|n=c

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