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Assuming you have sent an excellent Mini Cover Letter within the body of the Invite {300 limit characters}, and got accepted, then you need to continue with the Selling your Case. Why?
The Mini Cover Letter is not enough. Taking into consideration that now you are connected then you can use a Full LinkedIn Message to expand on the Skill Set you want to sell to the company. Not only that, but also remember that you are building a connection so the more interaction you have, the better.
Now, regarding the phrasing of the message, you have to be carefully distinguishing between a normal regular Connection and the one you may be making because you plan to apply to a Job Post published by the company. In this latter case, the Thanks Letter must to tailor to replace a Full Cover Letter.
References:
https://justinomora.net/node/60