I totally agree with Iain. If you had sex the day you got treatment, you weren't cured yet, and have just been passing this back and forth.
Both of you should get treatment now, and do not have sex for at least 7 days. Untreated chlamydia can cause damage to you especially, so this is really important.
And yes, definitely show this to your boyfriend.
Very likely. Have I got it correct that you.had unprotected sex on the day you started taking treatment and every day whilst you were on the treatment?? If the sex was unprotected then of course the treatment wouldn't work, why would it?? You can show your boyfriend this. Yes, the infection would absolutely keep going back and forth between you!