A. How do I prove the following sequence converges as $n$ goes to $\infty$ for any $c$, and how do I find the limit?
$ \begin{align} a_1 &=\frac{1}{c} \int _0^c\frac{x_1 }{1+x_1}\;dx_1 \\ \\ a_2 & =\frac{1}{c^2 } \int _0^c\int _0^c\frac{x_1 +x_2 }{2+x_1 +x_2 }\;dx_2\;dx_1 \\ \\ a_3 & =\frac{1}{c^3 } \int _0^c\int _0^c\int _0^c\frac{x_1 +x_2 +x_3 }{3+x_1 +x_2 +x_3 } \;dx_3\;dx_2\;dx_1 \end{align} $
and so on for $a_n$ $\dots$
B. Similarly, with this, where $f$ and $g$ are not polynomials [I verified the convergence numerically]:
$ \begin{align} a_1 &=\frac{1}{c} \int _0^c\frac{f(x_1) }{g(x_1)}\;dx_1 \\ \\ a_2 & =\frac{1}{c^2 } \int _0^c\int _0^c\frac{f(x_1) +f(x_2) }{g(x_1) +g(x_2) }\;dx_2\;dx_1 \\ \\ a_3 & =\frac{1}{c^3 } \int _0^c\int _0^c\int _0^c\frac{f(x_1) +f(x_2) +f(x_3) }{g(x_1) +g(x_2) +g(x_3) } \;dx_3\;dx_2\;dx_1 \end{align} $
and so on for $a_n$ $\dots$ (perhaps I need to include the very long definition of $f$ and $g$...?)