Will betta's fins grow back??

The best thing you can do is change at least 50% of the water every day. AFter he is healed, you can bring him back to the store to find a home. That way you would have a tank to take care of any more that need to heal up. There might be some scars, but they can heal quite well.
 
i agree with 50% water changes per day minimum, especially on an unfiltered tank and definitely once you have the new filter cycling. i wouldn't heat a tank that small, unless you are able to get the smallest preset heater rated for tiny tanks. if your room temperature stays stable and isn't too cool i wouldn't worry about heating a smaller tank. i had bettas in bowls ranging from 1 to 3 gallons and didn't heat any of them and the fish were fine.

his fins may grow in paler than what they were before, and you will see a white, sort of transparent, edge to them as they grow back. betta fins are actually only a few cells thick in some places.
 
With regard to heating, BFM, I think you are right that you have to look at the size of the tank. But I have had an appropriate rated heater in 2 gallon tank that worked well. I guess it depends on what your house temp is. Mine downstairs is around 68-70 and would not be healthy for a betta IMO.
So, just make sure to get the right type of heater. 78-80 will make your betta much more active and the stability of the temp will help keep him from getting diseases and infections.
 
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