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Theory and ExamplesA sequence of rational numbers is
Chapter 10, Problem 100E(choose chapter or problem)
Problem 100E
Theory and Examples
A sequence of rational numbers is described as follows:
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xn and yn be, respectively, the numerator and the denominator of the nth fraction rn = x n /y n .
a. Verify that and, more generally, that if or then
respectively.
b. The fractions rn = x n /y n approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that and that y n is not less than n.)
Questions & Answers
QUESTION:
Problem 100E
Theory and Examples
A sequence of rational numbers is described as follows:
Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xn and yn be, respectively, the numerator and the denominator of the nth fraction rn = x n /y n .
a. Verify that and, more generally, that if or then
respectively.
b. The fractions rn = x n /y n approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that and that y n is not less than n.)
ANSWER:
SOLUTION
Step 1 of 3
Here, we are given a sequence and are asked to verify the following.
The given sequence is
It is given that the nth fraction is of the form