   Chapter 11.10, Problem 47E

Chapter
Section
Textbook Problem

# Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f?47. f(x) = xe−x

To determine

To find: The Maclaurin series for f(x) and also the radius of the convergence and notice the relationship between Taylor polynomials and f(x) :To sketch the Maclaurin series for f(x) on the graph and also the first few Taylor polynomials on the same screen.

Explanation

The Maclaurin series is f(x)=n=1(1)n1xn(n1)! and radius of convergence is R= .

Given:

The function is xex .

Result used:

The expansion of ex is n=0xnn! . (1)

The radius of convergence of ex is .

(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 .

Calculation:

Consider the given function f(x)=xex .

Substitute x for x,

ex=n=0(x)nn!=n=0(1)nxnn!

Multiply by x on both sides

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