To solve the expression $1 - \frac{1}{1 + \frac{3}{4 + \frac{1}{2 + \frac{1}{7}}}}$, we'll proceed step by step, starting from the innermost fraction and working our way out.

Step 1: Simplify the innermost fraction

First, simplify the expression inside the innermost fraction:

$\frac{1}{7} \approx 0.142857$

Step 2: Simplify the next fraction

Now, add this to 2:

$2 + \frac{1}{7} = 2 + 0.142857 = 2.142857$

Step 3: Simplify the next fraction

Next, calculate the fraction with 3 in the numerator:

[ \frac{1}{2.