I did a short C code and, unless i made an error, it seems to diverge. So maybe you just need to prove it diverges
Edit: no it doesnt, my mistake, it seems to converge to 3.
#include < stdio.h>
int main()
{
float sum = 0, i = 0, u_n = 1.0/3.0;
for (i; i < 10; i++)
{...