The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e). $$ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots}}} $$ b. Evaluate the first five terms of the sequence $$ \left\{a_{n}\right\} $$. $$ a_{0}=2, a_{1}=\square, a_{2}=\square, a_{3}=\square, a_{4}=\square $$ (Type integers or decimals rounded to three decimal places as needed.)

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To evaluate the continued fraction $$ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots}}} $$, we can define the sequence $$ \{a_n\} $$ as follows:

- $$ a_0 = 2 $$
- $$ a_1 = 2 + \frac{1}{a_0} $$
- $$ a_2 = 2 + \frac{1}{a_1} $$
- $$ a_3 = 2 + \frac{1}{a_2} $$
- $$ a_4 = 2 + \frac{1}{a_3} $$

Now, let's calculate each term step by step.

1. **Calculate $$ a_1 $$**:
   $$
   a_1 = 2 + \frac{1}{a_0} = 2 + \frac{1}{2} = 2 + 0.5 = 2.5
   $$

2. **Calculate $$ a_2 $$**:
   $$
   a_2 = 2 + \frac{1}{a_1} = 2 + \frac{1}{2.5} = 2 + 0.4 = 2.4
   $$

3. **Calculate $$ a_3 $$**:
   $$
   a_3 = 2 + \frac{1}{a_2} = 2 + \frac{1}{2.4} \approx 2 + 0.41667 \approx 2.41667
   $$

4. **Calculate $$ a_4 $$**:
   $$
   a_4 = 2 + \frac{1}{a_3} = 2 + \frac{1}{2.41667} \approx 2 + 0.41322 \approx 2.41322
   $$

Now, rounding to three decimal places, we have:

- $$ a_0 = 2.000 $$
- $$ a_1 = 2.500 $$
- $$ a_2 = 2.400 $$
- $$ a_3 \approx 2.417 $$
- $$ a_4 \approx 2.413 $$

Thus, the first five terms of the sequence $$ \{a_n\} $$ are:

$$
a_{0} = 2.000, \quad a_{1} = 2.500, \quad a_{2} = 2.400, \quad a_{3} = 2.417, \quad a_{4} = 2.413
$$
$$ a_{0} = 2.000, \quad a_{1} = 2.500, \quad a_{2} = 2.400, \quad a_{3} = 2.417, \quad a_{4} = 2.413 $$

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