Have not check your whole calculation, but one reminder for you from the first line:

You have correctly stated that \( n \geq x \) in the summation, as required in a Binomial model. Note that you are calculating the pmf

\( \Pr\{X = x\} \)

and from the whole calculation process \( x \) is fixed from the very beginning. (And if this calculation results holds for all support point of \( X \), then you have finished calculating the whole pmf of \( X \))

For each fixed \( x \), we have

\( \Pr\{N = n, X = x\} = 0 \) when \( n < x \)

Therefore all the initial terms from \( n = 0 \) to \( n = x - 1 \) vanished, and you are actually summing from \( n = x \).