Sure! Let's solve the problem step-by-step.
Given the expression:
[tex]\[ 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \][/tex]
where [tex]\( x = 2011 \)[/tex].
First, let's start from the innermost fraction:
[tex]\[ \frac{1}{x} \][/tex]
Substituting [tex]\( x = 2011 \)[/tex],
[tex]\[ \frac{1}{2011} = 0.0004972650422675286 \][/tex]
Next, we substitute this value into the next layer:
[tex]\[ 1 + \frac{1}{\frac{1}{2011}} \][/tex]
which simplifies to:
[tex]\[ 1 + 2011 = 2012.0000000000002 \][/tex]
Now, we take the reciprocal of this result for the next layer:
[tex]\[ \frac{1}{2012.0000000000002} = 0.0004970178926441351 \][/tex]
Lastly, we add 1 to this value:
[tex]\[ 1 + 0.0004970178926441351 = 1.0004970178926442 \][/tex]
So, the value of the given expression [tex]\( 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}} \)[/tex] when [tex]\( x = 2011 \)[/tex] is:
[tex]\[ \boxed{1.0004970178926442} \][/tex]
