The first Dragon's Den series broadcast in 2005 was Series 1.
The British programme premiered with Series 1 on BBC Two in January 2005. This opening series introduced viewers to the UK version of the format, its original investor panel, and the basic structure of entrepreneurs pitching businesses in return for potential investment.
Series numbering is useful because the programme has continued across many years and has repeatedly changed its Dragons, presenters, and production details. Series 1 means the inaugural British run, not the first episode of a particular Dragon or the first international adaptation of the format.
The Japanese source programme and later overseas versions have their own histories and numbering systems. They should not be combined with the BBC's sequence. In the British programme, the 2005 launch is identified as Series 1, making that title the canonical answer to the question.