jmc

Consider the term $\frac{1}{x_{k-1}+1}.$ We can multiply the numerator and denominator by $x_{k-1},$ to get $$\frac{x_{k-1}}{x_{k-1}^2+x_{k-1}}=\frac{x_{k-1}}{x_k}.$$To get the sum to telescope, we can multiply the numerator and denominator again by $x_{k-1}$: $$\frac{x_{k-1}^2}{x_{k-1}{x}_k}=\frac{x_k-x_{k-1}}{x_{k-1}{x}_k}=\frac{1}{x_{k-1}}-\frac{1}{x_k}.$$Hence, $$\begin{array}{rl}\frac{1}{x_1+1}+\frac{1}{x_2+1}+\frac{1}{x_3+1}+\cdots & =\left(\frac{1}{x_1}-\frac{1}{x_2}\right)+\left(\frac{1}{x_2}-\frac{1}{x_3}\right)+\left(\frac{1}{x_3}-\frac{1}{x_4}\right)+\cdots \\ & =\frac{1}{x_1}=\boxed{\frac{1}{115}}.\end{array}$$