That is a really good question and I can see why are you in that dilemma. I think the question that you have to ask yourself is what you want to get in return for sending that birthday card? Are there any intentions behind sending that card? Do you want him to read it and then get him to somehow get back in contact with you. Is it a way in which you can feel like you can apologize for something?
All I know from my past experience is that when I broke up with my ex (been together for about 3 years) about a month later it was Christmas. I still hadn't let go of him in my head yet and so I went to his place to drop off some flowers for his whole family.
The truth of the matter was .. was that I was trying to manipulate the situation by giving him a gift in hope that he would give me something in return. SO I think if you do this, this will only make him feel a little guilty that he should do something. In a way its a little like still trying to hold on to that hope that you guys are completely separated.
i know its hard, but by not sending him the card, you will find that you have the strength to finally move on..
All the best
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