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Coenagrion puella
It needs some experience to tell the little blue damselflies apart, and even then it can be very hard. The one on your photo is a male, which makes it easier.
In some areas of Europe it is undoable to tell females of different Coenagrion species apart. Anyway, a (similar) site like http://observado.org has a number of volunteer specialists checking all observations. Here I just found your photo accidentally (and also my experience is nothing compared to that of the observado people).
Thank you very much :)
I have a book about danish insects, and as you can tell the drawings must be very poor. It is so difficult to tell any species from it, so I am very glad you could help me :)
It is not Enallagma cyathigerum, that shows on the thorax: this one has a second small stripe there, which is missing in Enallagma cyathigerum.
The segments 2-5 are exactly right for Coenagrion puella and not for Coenagrion pulchellum. Segment 2 (S2) has a U-shape, not a Y-shape (as C. pulchellum mostly has). S3-S5 look identical with a small black part. In C. pulchellum S3-S5 are not identical and usually have more black.
I think it look more like Enallagma Cyathigerum or Coenagrion Pulchellum because of the stribes...