   Chapter 11.10, Problem 3E

Chapter
Section
Textbook Problem

# If f(n)(0) = (n + 1)! for n = 0, 1, 2, …, find the Maclaurin series for f and its radius of convergence.

To determine

To find: The Maclaurin series for f and its radius of the convergence.

Explanation

Given:

The nth derivative of the function f(x) at the point 0 is, fn(0)=(n+1)! for n=1,2,3, .

Result used:

(1) The expansion of the Maclaurin series f(x)=n=0f(n)(0)n! is,

f(0)+f01!x+f(0)2!x2+f(0)3!x3+ (1)

(2) The Ratio Test:

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

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

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an

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