serve out (something)
To complete the required amount of time for some service. The senator announced that he would not be seeking reelection, but that he would serve out the rest of his time in office. Because of good behavior, the inmate was allowed to serve out the rest of her prison sentence in minimum security.
serve something out
to carry out one's duty or responsibility for the whole time, all the way to the end. She was unable to serve her term out. The convict served out his sentence in solitary confinement.
serve out
v. To complete some period of service: The criminal served out his sentence in a minimum security prison. After she serves out her term as governor, she will probably run for the Senate.