"arranged or disposed in a rank or row; forming part of a series; coming in regular succession," 1840, from series + -al (1). French sérial also seems to originate around 1840. Popularized in English in reference to Dickens' novels, which were first published over time in periodicals (as opposed to all at once in a book). The word was found to be useful to English and from the 1850s it was given wide application.
Serial number, indicating position in a series, first recorded 1866, originally of papers, packages, etc.; of soldiers from 1918. Serial killer is first attested 1981 (in relation to John Wayne Gacy and Ted Bundy), though serial had been used in connection with murders since the early 1960s. Related: Serially.
1660s, "bond, link, interdependence between members of a series or group; means of communication," from Latin nexus "that which ties or binds together," past participle of nectere "to bind," from PIE root *ned- "to bind, tie."
"interruption of continuity, separation of parts which form a connected series," 1610s, from French discontinuation (14c.), from Medieval Latin discontinuationem (nominative discontinuatio), noun of action from past-participle stem of discontinuare (see discontinue (v.)).