The answer of J. Arthur is correct. I know the word "continue" has confused you. Both of them indicate continuity of the action. Somehow the same but not completely. I know since we do not usually use this tense in Persian language It may sound confusing. Continue to expand ادامه دادن به گسترده کردن یا ادامه دادن به گسترش چیزی Continue expanding یعنی عمل گسترش شروع شده و ادامه دار هم هست و تو به این عمل که از قبل شروع شده و استمرار داره ادامه میدی In fact, the second sentence is just for more emphasizing the act of expansion.

Continue to expand: You continue this expansion. You are not necessarily doing it right now, but it is something you will continue to do. It can be in different phases. Continue expanding: You are now in the process of expanding, and will continue to do so up untill a certain point (for example when the work is done)

The answer would depend upon the context so that we know what definition/use of expanding is being referred to. 1 Expand means to make or become larger or more extensive. 2 in physics it means to undergo a continuous change. 3 it means to give a fuller account of something 4 it means for a person to become more mature, less reserved in their character, nature or behaviour. 4 May be British English only I am not sure. You will have to ask some Americans. so although they are the same: "continue to expand" you can say a sentence like this "The universe will continue to expand" That is the theory. Going to expand: you can say this also "the Universe is going to continue to expand" you are emphasising it will not stop that is the theory. For a person: Going to expand: "They are going to expand for the rest of their lives" = you think fell the person will continue to mature, although people do not often use the word expand in this scenario, they use the words grow/mature Continue Expanding: "they just continue expanding, every time I meet them they have grown more mature" For an account of something: Referred to as "EXPANDING ON" Continue to expand" "Why do you continue to expand on everything I tell you constantly giving 5 minute updates" continue expanding: continue expanding: "Why do you continue expanding on everything I tell you constantly giving 5 minute updates" continue expanding: "Will the universe continue expanding?" will it ever stop. continue to expand: "Will the universe continue to expand?" will it ever stop. So basically it is the perspective from which you want to ask the question or view the expansion from that determines which version to use, it doesn't affect the meaning

Thank you Kevin

They are the same, but there might be some situations where one is better than the other based on the style of writing. In spoken English, they are interchangeable. I could continue to expand this answer, but if I continue expanding it, it might now be any more helpful than it is now.