"room temperature" is fairly generic :/ you will need a heater for your betta.
I would recommend a 25w adjustable heater.
For his current situation you need to do a complete water change. which means taking the fish out in a temporary container. (with a lid or some kind of cover!) then dumping all the water.
wash everything with hot water.
add new water (this water should be the same temperature as that of the old
place fish back in.
now you just did a 100% water change. in addition to that everyweek, you also need to do a 50% water change half week.
Otherwise your betta will be swimming in accumulated toxins and fall prone to disease again.
The black tail is fin rot, clean warm water should cure it.
Here is a shopping list for you :)
Highly highly recommended items:
Adjustable 25w submersible heater
stresscoat/prime water conditioner (prime is better)
turkey baster (to take out uneaten food)
airline (to siphon)
1gallon bottles (to make water changes easier)
frozen blood worms ( for variation in diet)
freeze dried blood worms (if you cant purchase the above)