TITLE: Find the limit of $f_{n}(1)$ if $f_{n}(x)=\int^{x}_{0}f_{n-1}(y)\,dy$ for each $n$
QUESTION [8 upvotes]: Consider the sequence $(f_n)$ defined by
  $$
f_{0}(x)=\frac{1}{(1+x)^3}\quad
f_{n}(x)=\int^{x}_{0}f_{n-1}(y)\;\text{dy},\ n\ge1.$$
  Find 
  $$\lim_{n\to \infty}f_{n}(1).$$

I computed $f_{1}(x)$, $f_{2}(x)$, $f_{3}(x)$, but I couldn’t get it that way.

REPLY [3 votes]: If I'm not mistaking you can transform your integral into
$$f_n(x)=f^{(-n)}(x)=\int_{0}^x\int_{0}^{a_1}\int_0^{a_2}\cdots\int_0^{a_{n-1}}f_0(a_n)da_nda_{n-1}\cdots da_2da_1=\\=\frac{1}{(n-1)!}\int_{0}^{x} (x-t)^{n-1}\frac{1}{(1+t)^3}$$
by using the Cauchy formula for repeated integration,the integral converges for $x=1$ and $\frac{1}{(n-1)!}$ diverges (if I'm not mistaking)