Geometric Series Formula

The man discovers the following mathematical solution:

511000 = 1000 * (2^n - 1) / (2 - 1)

Mathematical steps

511000 = 1000 * (2^n - 1)

511 = 2^n - 1

512 = 2^n

log2 512 = n

n = 9

Conclusion

The largest number of consecutive losses is 9 rounds.

Probability of Bankruptcy

The chance he gets bankrupt is 1/2^9 = 1/512