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

Question

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

UpStudy AI · Accepted Answer

1. Let the infinite continued fraction be denoted by $$ C $$:
   $$
   C = 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \cdots}}}.
   $$

2. We define a sequence $$ \{a_n\} $$ where each term is the finite continued fraction up to $$ n $$ layers. For example:
   - $$ a_1 = 2 $$
   - $$ a_2 = 2 + \frac{1}{2} $$
   - $$ a_3 = 2 + \frac{1}{2 + \frac{1}{2}} $$
   - and so on.

3. As $$ n $$ increases, $$ a_n $$ becomes a closer approximation to the infinite process. That is, for large $$ n $$, the value of $$ a_n $$ is very close to $$ C $$.

4. If the limit $$ \lim_{n\to\infty} a_n $$ exists, it represents the value that the finite approximations are approaching. Thus, by the definition of convergence, the infinite continued fraction is equal to this limit.

5. Therefore, the answer is:

B. $$ a_{n} $$ is very close to the value of the continued fraction for large values of $$ n $$. So, it follows that the value of the continued fraction is equal to $$ \lim_{n\to\infty} a_n $$.
The value of the continued fraction is equal to the limit of the sequence $$ a_n $$ as $$ n $$ approaches infinity.

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