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Hi

I am sure you can do the same as me. Or can you?

--- Is "or" necessary? Is it correct?
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NewguestIs "or" necessary? Is it correct?
It's necessary and correct. The or is what makes the question an expression of doubt and to some extent a challenge.

I am sure you can ....
Or can you? = Or am I mistaken, and you really cannot?
Or can you? = But maybe I'm mistaken, and you are not as capable a person as I thought.

CJ
What about:

You can't swim, can you?

Or maybe: You can't swim, or can you?
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NewguestI am sure you can do the same as me. Or can you?
This one is a bit subtle. The OR makes a lot of difference. I will describe both versions separately.
"I'm sure you can do the same as me. Can you?"
This is not idiomatic. It would mean the same as "I'm sure you can do the same. Don't you agree?". I'm making the statement, then asking whether he agrees or not, and "DON'T you agree?" implies that you expect him to agree.
"I'm sure you can do the same as me. Or can you?"
In this one, the "Or can you?" is a second thought.
You're saying you think he can do the same, then you pause and think again, and wonder whether you're right or not, so you add "Or can you?"
It would normally be written "I'm sure you can do the same... Or can you?" where the "..." indicates a pause, while you're thinking.
KrisBlueNZ"I'm sure you can do the same as me. Can you?"

Actually I wanted to write: I'm sure you can do the same as me. Can't you?
NewguestI'm sure you can do the same as me. Can't you?
This is very similar, but subtly different again.
You're saying "I'm sure you can", then pausing (the full stop, or period, ".", indicates the pause). Then you have "second thoughts" - you're not sure any more. So you say "Can't you?", which means "I think you can, or I thought you could, but maybe you can't, and if you can't, say so."
Then the other person's reply would be "yes" if he can, and "no" if he can't, which is kind of the opposite of what you might expect. If he agrees that he can, he says "yes", but he's disagreeing with "Can't you?". So he is answering the first part of the statement, with "yes, I can" or "no, I can't". The second part, "Can't you", is not the question he is answering. If he says "yes", he's not saying "Yes I CAN'T", he's saying "Yes I CAN".
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correct answer : can't you ?
explanation : the sentence above is a question tag. if the sentence is positive, the question tag must be in negative.
I am sure you can do the same as me. can't you ?