Incorrect spelling, explanation: the noun siege comes from Old French, that’s why its structure reminds us of the French language. When writing down this word, we must bear in mind its origin and etymology. We can clearly see that after -s should be -i, then -e. Keeping this in mind, we will not make a spelling mistake. Seige is not correct.

Correct spelling, explanation: if we are not sure how to write a given word (especially when there are two vowels next to each other), we should always check its etymology. In the case of the word siege, it is easy to see that it cannot be spelled seige.

Definition of siege:
noun, a military act of surrounding a city or base
During the siege people decided to gather food and supplies.
I hope this siege will not last long for the sake of the city’s inhabitants.