I assume that you are asking about the sort of convergence in which the economies of various nations catch up to that of a nation that has been ahead. This can be good for both the countries that catch up and the country that was previously ahead. The benefits for the countries that catch up are fairly obvious. They become richer and their people come to have a higher standard of living. But this can also help the former leader. As the other countries get richer, that country can hope to sell more things like luxury goods to the people in those countries who could not previously afford them. There might also be benefits in things like reduced migration (some of it illegal) to the richer country. In these ways, convergence can help even the country that is losing its dominant position.