How to tell if this sequence converges?

Question

I have a function $f(x) = \frac{1}{2}x^4$, where $x = 1$ initially. Every time step, the function is updated with the rule $x = x - \alpha * dx$. I want to know if the function will converge to 0 at $t = \infty$ and if so, for what ranges of the scalar $\alpha$.

I know $dx = 2x^3$ and this problem sort of reminds me of a combination of geometric/arithmetic series, but I'm not sure how to go about it. I wrote a Python script to try out different values, but I was hoping to have a more analytic solution.

def dx(x):
  return 2 * x ** 3

def fx(x):
  return 0.5 * x ** 4

i = 0
x = 1
alpha = 0.9
while i < 20:
  x -= alpha * dx(x)
  print("{}: {}".format(i, x))
  i += 1

I would really appreciate some advice on how to approach this problem.