Simplify 1+1/(1+1/(1+1/x))

$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}$

Simplify terms.

Simplify each term.

Simplify the denominator.

Write $1$ as a fraction with a common denominator.

$1 + \frac{1}{1 + \frac{1}{\frac{x}{x} + \frac{1}{x}}}$

Combine the numerators over the common denominator.

$1 + \frac{1}{1 + \frac{1}{\frac{x + 1}{x}}}$

Simplify the denominator.

Write $1$ as a fraction with a common denominator.

$1 + \frac{1}{\frac{\frac{x + 1}{x}}{\frac{x + 1}{x}} + \frac{1}{\frac{x + 1}{x}}}$

Combine the numerators over the common denominator.

$1 + \frac{1}{\frac{\frac{x + 1}{x} + 1}{\frac{x + 1}{x}}}$

Rewrite $\frac{\frac{x + 1}{x} + 1}{\frac{x + 1}{x}}$ in a factored form.

Write $1$ as a fraction with a common denominator.

$1 + \frac{1}{\frac{\frac{x + 1}{x} + \frac{x}{x}}{\frac{x + 1}{x}}}$

Combine the numerators over the common denominator.

$1 + \frac{1}{\frac{\frac{x + 1 + x}{x}}{\frac{x + 1}{x}}}$

Reorder terms.

$1 + \frac{1}{\frac{\frac{x + x + 1}{x}}{\frac{x + 1}{x}}}$

Add $x$ and $x$ .

$1 + \frac{1}{\frac{\frac{2 x + 1}{x}}{\frac{x + 1}{x}}}$

Multiply the numerator by the reciprocal of the denominator.

$1 + \frac{1}{\frac{2 x + 1}{x} \cdot \frac{x}{x + 1}}$

Cancel the common factor of $x$ .

Cancel the common factor.

$1 + \frac{1}{\frac{2 x + 1}{x} \cdot \frac{x}{x + 1}}$

Rewrite the expression.

$1 + \frac{1}{(2 x + 1) \frac{1}{x + 1}}$

Multiply $2 x + 1$ by $\frac{1}{x + 1}$ .

$1 + \frac{1}{\frac{2 x + 1}{x + 1}}$

Multiply the numerator by the reciprocal of the denominator.

$1 + 1 \frac{x + 1}{2 x + 1}$

Multiply $\frac{x + 1}{2 x + 1}$ by $1$ .

$1 + \frac{x + 1}{2 x + 1}$

Combine into one fraction.

Write $1$ as a fraction with a common denominator.

$\frac{2 x + 1}{2 x + 1} + \frac{x + 1}{2 x + 1}$

Combine the numerators over the common denominator.

$\frac{2 x + 1 + x + 1}{2 x + 1}$

Simplify the numerator.

Add $2 x$ and $x$ .

$\frac{3 x + 1 + 1}{2 x + 1}$

Add $1$ and $1$ .

$\frac{3 x + 2}{2 x + 1}$