Explanation Given: The sequence is a n = ( − 1 ) n n n + 1 . Calculation: Obtain the first 20 terms of the sequence. n n n + 1 a n = ( − 1 ) n n n + 1 1 1 2 = 0.5000 − 0.5000 2 2 3 = 0.6667 0.6667 3 3 4 = 0 .75 00 − 0 .75 00 4 4 5 = 0.8000 0.8000 5 5 6 = 0 .8333 − 0 .8333 6 6 7 = 0 .8571 0 .8571 7 7 8 = 0 .875 0 − 0 .875 0 8 8 9 = 0 .8889 0 .8889 9 9 10 = 0.9000 − 0.9000 10 10 11 = 0 .9091 0 .9091 11 11 12 = 0 .9167 − 0 .9167 12 12 13 = 0 .9231 0 .9231 13 13 14 = 0 .9286 − 0 .9286 14 14 15 = 0 .9333 0 .9333 15 15 16 = 0 .9375 − 0 .9375 16 16 17 = 0 .9412 0 .9412 17 17 18 = 0 .9444 − 0 .9444 18 18 19 = 0 .9474 0 .9474 19 19 20 = 0 .95 00 − 0 .95 00 20 20 21 = 0 .9524 0 .9524 Plot the points ( n , a n ) , for n = 1 , 2 , ...20 on the graph as shown below in Figure 1. From the graph, it is observed that the sequence is divergent because the sequence oscillates between −1 and 1. lim n → ∞ a n = lim n → ∞ ( ( − 1 ) n n n + 1 ) . Use the subsequence n = 2 m in equation (1). lim n → ∞ ( ( − 1 ) n n n + 1 ) = lim m → ∞ ( ( − 1 ) 2 m ⋅ 2 m 2 m + 1 ) = lim m → ∞ ( ( ( − 1 ) 2 ) m ⋅ 2 m 2 m + 1 ) = lim m → ∞ ( 1 m ⋅ 2 m 2 m + 1 ) = lim m → ∞ ( 2 m 2 m + 1 ) Divide the numerator and denominator by the highest power. lim m → ∞ ( ( − 1 ) 2 m ⋅ 2 m 2 m + 1 ) = lim m → ∞ ( 2 m m 2 m + 1 m ) = lim m → ∞ ( 2 2 + 1 m ) = lim m → ∞ ( 2 ) lim m → ∞ ( 2 + 1 m ) = lim m → ∞ ( 2 ) lim m → ∞ ( 2 ) + lim m → ∞ ( 1 m ) Apply infinity property lim n → ∞ ( k n a ) = 0 and simplify the terms

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 8.2, Problem 36E

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To find: The limit of the sequence or determine if limit does not exist; Verify the result with a graphing utility.

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Chapter 8.2, Problem 35E

Chapter 8.2, Problem 37E

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(#1) Consider the solid bounded below by z = x² and above by z = 4-y². If we were to project this solid down onto the xy-plane, you should be able to use algebra to determine the 2D region R in the xy-plane for the purposes of integration. Which ONE of these limite of integration would correctly describe R? (a) y: x24x: -22 - (b) y: 22 x: 04-y² (c) y: -√√4-x2. →√√4x²x: −2 → 2 (d) z: 24-y² y: -2 → 2 (e) None of the above

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Chapter 8 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 8.1 - Define sequence and give an example.Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Given the series k=1k, evaluate the first four...Ch. 8.1 - The terms of a sequence of partial sums are...Ch. 8.1 - Consider the infinite series k=11k. Evaluate the...Ch. 8.1 - Explicit formulas Write the first four terms of...Ch. 8.1 - Explicit formulas Write the first four terms of...

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The limit of the sequence or determine if limit does not exist; Verify the result with a graphing utility.

Solution Summary: The author explains how to find the sequence's limit or determine if it doesn't exist.

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To find: The limit of the sequence or determine if limit does not exist; Verify the result with a graphing utility.

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Chapter 8 Solutions