Pregunta
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The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e).
Explain why the value of the expression can be interpreted as , provided the limit exists. Choose the correct answer.
A. is always less than the continued fraction. So, if exists, it follows that the value of the continued fraction is equal to .
B. is very close to the value of the continued fraction for large values of . So, it follows that the value of the continued fraction is equal to .
C. is a constant multiple of the continued fraction for large values of . So, it follows that the value of the continued fraction is equal to .
D. is always greater than the continued fraction. So, if exists, it follows that the value of the continued fraction is equal to .

Ask by Warren Hills. in the United States
Mar 15,2025

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The value of the continued fraction is equal to the limit of the sequence as approaches infinity.

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Beyond the Answer

The magical world of continued fractions shows us a clever way of rewriting numbers! You can think of the expression as a series of nested fractions, and as we keep digging deeper, we approach a certain value. When we say , we’re essentially saying that as you add more and more layers, you get closer and closer to this magical number.
For this specific continued fraction, we can see that as increases, the approximations change but stabilize around a number. The correct choice here is B. As grows, becomes a better approximation, and that’s why we can deduce that the limit represents the true value of the entire expression. How neat is that?

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