From what I have read about it French Tarragon does benefit from a cool weather period but it is sensitive to more than a very light frost.
It can be grown in partial shade and likes soil 6.5 to 7 ph. It does not like acidic soils. A light foliar feeding of seaweed or compost tea a couple of times during the growing season is all it needs. It is fairly pest and disease free and needs little water once established.
Flowers are sterile but plants can be easily divided by rhizomes or by placing a 6" to 8" cutting in moist sand for a about 4 weeks to allow root development.
I may be wrong, but it sounds like a plant that will be tolerant of the inevitable neglect that I will bestow upon it.
