Explanation Definition used: “The sequence { a n } converges to L provided the terms of a n can be made arbitrarily close to L by taking n sufficiently large. That is, { a n } has the unique limit L if given any ε > 0 , it is possible to find a positive integer N (depending only on ε ) such that | a n − L | < ε whenever n > N , which can be expressed as lim n → ∞ a n = L .” Proof: Consider the sequence a n = c n b n + 1 . Choose the positive integer N and given any ε > 0 . From the definition, there exist a real number n sufficiently large such that, | a n − c b | < ε whenever n > N . That is, | c n b n + 1 − c b | < ε whenever n > N . Solve and obtain the value of n . | b c n − b c n − c b ( b n + 1 ) | < ε for n > N | − c b ( b n + 1 ) | < ε c b ( b n + 1 ) < ε c < b ( b n + 1 ) ε for n > N Simplify further and compute N

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ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

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Chapter 8.2, Problem 73E

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To prove: The limit of the sequence {cnbn+1} is cb for real numbers b>0 and c>0 using the formal definition of the limit of the sequence.

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Chapter 8.2, Problem 72E

Chapter 8.2, Problem 74E

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To prove: The limit of the sequence { c n b n + 1 } is c b for real numbers b > 0 and c > 0 using the formal definition of the limit of the sequence.

Solution Summary: The author explains that the sequence leftcnbn+1right converges to L provided the terms of a_n can be made arbitrarily close to

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To prove: The limit of the sequence {cnbn+1} is cb for real numbers b>0 and c>0 using the formal definition of the limit of the sequence.

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Chapter 8 Solutions

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To prove: The limit of the sequence { c n b n + 1 } is c b for real numbers b &gt; 0 and c &gt; 0 using the formal definition of the limit of the sequence.

Solution Summary: The author explains that the sequence leftcnbn+1right converges to L provided the terms of a_n can be made arbitrarily close to

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To prove: The limit of the sequence {cnbn+1} is cb for real numbers b>0 and c>0 using the formal definition of the limit of the sequence.

Want to see the full answer?

Chapter 8 Solutions

Additional Engineering Textbook Solutions

To prove: The limit of the sequence { c n b n + 1 } is c b for real numbers b > 0 and c > 0 using the formal definition of the limit of the sequence.