Rantum on the island of Sylt and Rantrum near Husum would both be in Schleswig-Holstein and under Prussian, then German rule, when Jens Christian Clausen was growing up. Families there might well might identify as Danish.
Sophie Emilie Jurgens and Jens Christian Clausen are included in a family tree on Ancestry.com (perhaps yours?).
You posted this additional information in another Ancestry message board post: "Jens married Sophie Emilie Jürgens b. Sept 15, 1868 of Dunsum, Föhr. They had 5 children:
Olga Johanna b. Nov.25, 1891
Anna Catharina b. Aug. 20,1893
Jenny Wilhelmine b. Dec. 04, 1895
Gustav b. Dec. 25, 1897
Jürgen Wilhelm b. June 18, 1899
Sophie died May 05, 1905. Jens married the widow Auguste Wilhelmine Schugart geb. Backen. Jens died Feb. 05, 1939."
A photo posted on Flickr called "Clausen children of Dunsum, Fohr" has been commented on: "Olga Johanna is my grandmother. --Oluf Matzen" Another comment by him here:http://www.flickr.com/photos/62799230@N07/5720540650/
Perhaps adding this information can help volunteers who wish to help - I hope so.
A genealogy link in Schleswig-Holstein, with some interesting information about the area: http://www.rootdigger.de/