Hovedstadsregionen (now obsolete) was the name of the greater metropolitan region around and including Copenhagen

Copenhagen

Copenhagen is the capital and largest city of Denmark, with an urban population of 1,199,224 and a metropolitan population of 1,930,260 . With the completion of the transnational Øresund Bridge in 2000, Copenhagen has become the centre of the increasingly integrating Øresund Region...

 (København), Denmark

Denmark

Denmark is a Scandinavian country in Northern Europe. The countries of Denmark and Greenland, as well as the Faroe Islands, constitute the Kingdom of Denmark . It is the southernmost of the Nordic countries, southwest of Sweden and south of Norway, and bordered to the south by Germany. Denmark...

. It included the counties of Copenhagen, Frederiksborg and Roskilde.

It was replaced by Region Hovedstaden (Copenhagen Capital Region)

Region Hovedstaden

The Capital Region of Denmark is an administrative region of Denmark established on January 1, 2007 as part of the 2007 Danish Municipal Reform, which replaced the traditional counties with five larger regions. At the same time, smaller municipalities were merged into larger units, cutting the...

as an administrative region on January 1, 2007 as part of the 2007 Danish Municipal Reform.
However this region includes the island of Bornholm (in the Baltic Sea 100 nautical miles from Copenhagen) and excludes the south-west parts of greater Copenhagen. A more practical geographical term is the Copenhagen local transport area - the HUR area, which includes 2900 square kilometers of land and 1.881.984 inhabitants.