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The phrase "serve out" is correct and usable in written English.
It can be used to mean to complete a sentence or period of service, or to deliver something (e.g. a punishment). Example sentence: The soldier was required to serve out his full five-year term before he was eligible for promotion.
Dictionary
serve out
verb
To win a set, or by extension a match, by holding serve (winning a game as the server).
Exact(60)
Few manage to serve out a full 14-year term.
Newt Gingrich will serve out his full term.
First serve in, point won, first serve out, point lost.
He was ready to serve out the match.
An ace then a stinging serve out wide.
Murray dupes Verdasco with a sliced serve out wide.
A wonderful serve out wide makes it 40-0.
The linesman who called the serve out did not intervene.
Would he serve out his promised 10-year term?
A decent serve out wide from Li and Clijsters nets.
And an unreturned serve out wide secures the game.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com