Question

If x_0 != 0, show SUM n!(x_0)^n (n=0 to inf) diverges.

Answer 1

Write a_n for the nth term, a_n = n!(x_0)^n.

If the sum converges then
lim a_n+1/a_n < 1.

Hence if lim a_n+1/a_n >= 1, then the sum doesn't converge.

Show that lim a_n+1 / a_n is not less than 1.