   Chapter 11.6, Problem 45E

Chapter
Section
Textbook Problem

# (a) Show that ∑ n − 0 ∞ x n / n ! converges for all x. (b) Deduce that limn→∞ xn/n! = 0 for all x.

(a)

To determine

To show: The series n=0xnn! converges for all x.

Explanation

Given:

The series is n=0xnn!.

Result used: 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:

The given series n=1an=n=0xnn!.

Substitute, n=n+1, then an+1=xn+1(n+1)!

(b)

To determine

To deduce: limnxnn!=0 for all x.

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