The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e). \[ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots}}} \] Explain why the value of the expression can be interpreted as \( \lim _{n \rightarrow \infty} a_{n} \), provided the limit exists. Choose the correct answer. A. \( a_{n} \) is always less than the continued fraction. So, if \( \lim _{n \rightarrow \infty} a_{n} \) exists, it follows that the value of the continued fraction is equal to \( \lim _{n \rightarrow \infty} a_{n} \). 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 \( \lim _{n \rightarrow \infty} a_{n} \). C. \( a_{n} \) is a constant multiple of the continued fraction for large values of \( n \). So, it follows that the value of the continued fraction is equal to \( \lim _{n \rightarrow \infty} a_{n} \). D. \( a_{n} \) is always greater than the continued fraction. So, if \( \lim _{n \rightarrow \infty} a_{n} \) exists, it follows that the value of the continued fraction is equal to \( \lim _{n \rightarrow \infty} a_{n} \).
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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 \( \lim _{n \rightarrow \infty} a_{n} \), 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 \( n \) increases, the approximations \( a_{n} \) change but stabilize around a number. The correct choice here is B. As \( n \) grows, \( a_{n} \) 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?
