   Chapter 11.5, Problem 21E

Chapter
Section
Textbook Problem

# Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.21. ∑ n = 1 ∞ ( − 0.8 ) n n !

To determine

To sketch: The sequence of terms and the sequence of partial sums on the graph and then estimate the sum of the series correct to four decimal places.

Explanation

Given:

The series is n=1(0.8)nn!.

Here, the sequence of the term is an=(0.8)nn!.

Result used:

(1) Alternating Series Estimation Theorem:

“If s=(1)n1bn, where bn>0, is the sum of an alternating series that satisfies the conditions, bn+1bn and limnbn=0, then |Rn|=|ssn|bn+1.”

(2) The geometric series n=1arn1 is convergent if |r|<1 and divergent if |r|>1.

Calculation:

The nth term of the partial sum is sn=i=1nai.

Obtain the first 10 terms of the sequence and partial sums.

 n an=(−0.8)nn! sn=∑i=1nai 1 a1=−0.80000 s1=−0.800000 2 a2=0.32000 s2=−0.48000 3 a3=−0.0853¯ s3=−0.5653¯ 4 a4=0.01706 s4=−0.54826 5 a5=−0.00273 s5=−0.55099 6 a6=0.00364 s6=−0.55063 7 a7=−0.00004161 s7=−0.550674 8 a8=0.00000416 s8=−0.5506706 9 a9=−0

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