should I be concerned about my boyfriends ex?

Okay so I have a question I want different people's opinions. So my boyfriend has this ex who he used to be best friends with before they started dating and they dated on and off. When we met he had started talking to her again but a week after we met he stopped talking to her bc he realized he really liked me and then we started going out on dates. And after about two months of us going out on dates(but never being official bc I made it clear I wanted to date a bit before we decided to become official). Now we are official I saw that he has been texting his ex and I mentioned something and he was like yeah she just keeps texting me and I don't want to be mean and we used to be best friends. And I kinda brushed it off bc he wasn't texting her frequently. But now it seems like they're always texting and she's always asking for them to hang out but from what he's told me he has turned her down everytime. But the weirdest thing about it is that I'm pretty sure he hasn't told her that he has girlfriend. I trust him very much but I'm just wondering if I should be conerened about this or if I'm just over thinking things. Please give me some advice on what to do