Find the value of `2+1/(2+1/(2+1/(2+oo)))`

Explanation

To solve the expression \( 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \infty}}} \), we will first simplify the nested fractions step by step.

### Step 1: Understanding the Infinite Nesting
The expression \( 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \infty}}} \) suggests that as we go deeper into the nesting, the innermost part approaches a limit. We can denote the entire expression as \( x \):
\[ x = 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \infty}}} \]

### Step 2: Simplifying the Innermost Fraction
We can denote the innermost part as \( y \):
\[ y = 2 + \frac{1}{2 + \frac{1}{2 + \infty}} \]
As \( \frac{1}{2 + \infty} \) approaches \( 0 \), we can simplify \( y \) to:
\[ y = 2 + 0 = 2 \]

### Step 3: Substitute Back to the Next Level
Now we substitute \( y \) back into the expression for \( x \):
\[ x = 2 + \frac{1}{2 + \frac{1}{2}} \]

### Step 4: Simplifying Further
Next, we simplify \( \frac{1}{2 + \frac{1}{2}} \):
\[ 2 + \frac{1}{2} = 2 + 0.5 = 2.5 \]
Thus,
\[ \frac{1}{2 + \frac{1}{2}} = \frac{1}{2.5} = \frac{2}{5} \]

### Step 5: Final Calculation
Now substitute \( \frac{2}{5} \) back into the expression for \( x \):
\[ x = 2 + \frac{2}{5} \]
To add these, convert \( 2 \) to a fraction:
\[ 2 = \frac{10}{5} \]
Thus,
\[ x = \frac{10}{5} + \frac{2}{5} = \frac{12}{5} \]

### Final Result
The value of the expression \( 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \infty}}} \) is:
\[ \boxed{\frac{12}{5}} \]