Chapter 9.5, Problem 3E

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770

Chapter
Section

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770
Textbook Problem

Numerical and Graphical Analysis In Exercises 5-8, explore the Alternating Series Remainder.(a) Use a graphing utility to find the indicated partial sum S n and complete the table. n 1 2 3 4 5 6 7 8 9 10 S n (b) Use a graphing utility to graph the first 10 terms of the sequence of partial sums and a horizontal line representing the sum.(c) What pattern exists between the plot of the successive points in part (b) relative to the horizontal line representing the sum of the series? Do the distances between the successive points and the horizontal line increase or decrease?(d) Discuss the relationship between the answers in part (c) and the Alternating Series Remainder as given in Theorem 9.15. ∑ n = 1 ∞ ( − 1 ) n     1 n 2 = π 2 12 ,

To determine

To fill: To complete the table for the series n=1(1)n1n2=π212.

 n 1 2 3 4 5 6 7 8 9 10 Sn
Explanation

Consider the summation series,

Sn=n=1(1)n1n2 …...…...…...…...…...…...…...…...…...…...…...…...…... (1)

To complete the table, put n=1,2,3 in equation (1) that is,

Put n=1,

S1=n=11(1)n1(1)2=(1)111=11=1

Now, put n=2,

S2=n=12(1)n1(n)2=(1)11(1)2+(1)21(2)2=114=0.75

Now, put n=3,

S3=n=13(1)n1(n)2=(1)11(1)2+(1)21(2)2+(1)31(3)2=0.75+19=0.8611

Now, put n=4,

S4=n=14(1)n1(n)2=(1)11(1)2+(1)21(2)2+(1)31(3)2+(1)41(4)2=0.8611116=0.7986

Now, put n=5,

S5=n=15(1)n1(n)2=(1)11(1)2+(1)21(2)2+(1)31(3)2+(1)41(4)2+(1)51(5)2=0.7986+125=0.8386

Now, put n=6,

S6=n=16(1)n1(n)2=[(1)11(1)2+(1)21(2)2+(1)31(3)2+(1)41(4)2+(1)51(5)2+(1)61(6)2]=0.8386136=0.8108

Now, put n=7,

S7=n=17(1)n1(n)2=[(1)11(1)2+(1)21(2)2+(1)31(3)2+(1)41(4)2+(1)51(5)2+(1)61(6)2+(1)71(7)2]=0

(b)

To determine

To graph: The first ten terms of the sequence n=1(1)n1n2 and a horizontal line that represents the sum.

(c)

To determine
The pattern of the plot of successive points relative to the horizontal line that represents the sum of series, Sn=n=1(1)n1n2. And to determine the distance between successive points and the horizontal line is decreasing or increasing.

(d)

To determine
The relationship between the answer in part (c) and the series Sn=n=1(1)n1n2

 n 1 2 3 4 5 6 7 8 9 10 Sn 1 0.75 0.8611 0.7986 0.8386 0.8108 0.8312 0.8156 0.828 0.818

Graph of series is given by:

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