The following expression, where the process continues indefinitely, is called a continued
fraction. Complete parts (a) through (e).
d. Assuming the limit exists, use the fact that to determine the
limit exactly. Select the correct choice and, if necessary, fill in the answer box to
complete your choice.
A. The limit is . (Type an exact answer, using radicals as needed.)
B. The limit does not exist.

Ask by Johnson Cross. 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).
d. Assuming the limit exists, use the fact that to determine the
limit exactly. Select the correct choice and, if necessary, fill in the answer box to
complete your choice.
A. The limit is . (Type an exact answer, using radicals as needed.)
B. The limit does not exist.

Ask by Johnson Cross. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

Let $$ L $$ be the limit of the infinite continued fraction. Since the process continues indefinitely, the part of the fraction “after the first step” is identical in structure to the whole fraction. Hence, if $$ a_{n+1} $$ represents the remaining fraction, we have

$$
L = \frac{1}{1+L}.
$$

Multiplying both sides by $$ 1+L $$ gives

$$
L(1+L) = 1.
$$

This expands to

$$
L^2 + L - 1 = 0.
$$

Using the quadratic formula,

$$
L = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2},
$$

and since $$ L $$ must be positive, we have

$$
L = \frac{-1 + \sqrt{5}}{2} = \frac{\sqrt{5} - 1}{2}.
$$

Select choice A: The limit is $$ \frac{\sqrt{5} - 1}{2} $$.
The limit is $$ \frac{\sqrt{5} - 1}{2} $$.

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