Emperor Trajan has a son

Emperor Trajan was an interesting ruler, who led some early reform of the Roman Empire. He had no natural children but was succeeded by his adopted son, Hadrian. My question is this, if Trajan and his wife had had a son, who grew up to be a reasonably good administrator, and somewhat interested more in consolidated current gains, what consequences could this have? Would this son naturally ascend to the imperial throne?
 
This son would succeed Trajan. He probably try integrate his father's conquests to Roman empire and secure current borders, speciality in Britannia and Germania. Another question is how long dynasty would last. Probably not many generations. Romans emperors were pretty good mess everything.
 
This son would succeed Trajan. He probably try integrate his father's conquests to Roman empire and secure current borders, speciality in Britannia and Germania. Another question is how long dynasty would last. Probably not many generations. Romans emperors were pretty good mess everything.

Hmm this is true, but if he is more successful in securing the current borders, that would change quite a bit
 
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