From Förberedande kurs i matematik 1

When we come across large and complicated expressions, we have to work step by step;

as a first goal, we can multiply the top and bottom of the fraction

\displaystyle \frac{1}{1+\frac{1}{1+x}}

by \displaystyle 1+x, so as to reduce it to an expression having one fraction sign:

\displaystyle \begin{align} & \frac{1}{1+\frac{1}{1+\frac{1}{1+x}}}=\frac{1}{1+\frac{1}{1+\frac{1}{1+x}}\centerdot \frac{1+x}{1+x}}=\frac{1}{1+\frac{1+x}{\left( 1+\frac{1}{1+x} \right)\left( 1+x \right)}} \\ & \\ & =\frac{1}{1+\frac{1+x}{1+x+\frac{1+x}{1+x}}}=\frac{1}{1+\frac{1+x}{1+x+1}}=\frac{1}{1+\frac{x+1}{x+2}} \\ \end{align}

The next step is to multiply the top and bottom of our new expression by \displaystyle x+2, so as to obtain the final answer,

\displaystyle \begin{align} & \frac{1}{1+\frac{x+1}{x+2}}\centerdot \frac{x+2}{x+2}=\frac{x+2}{\left( 1+\frac{x+1}{x+2} \right)\left( x+2 \right)}=\frac{x+2}{x+2+\frac{x+1}{x+2}\left( x+2 \right)} \\ & \\ & \frac{x+2}{x+2+x+1}=\frac{x+2}{2x+3} \\ & \\ \end{align}