Nope you are right on the mark in assuming this is not typical behavior. I see 3 major factor contributing to your betta's lethargy:
1) Small tank.
To start, you are going to need a tank a minimum
of 4 times the size of your current one (2 gallons) but if this is your first betta you will want to go with a 5 gallon tank. Why? Here is a quote from a similar thread I posted a while ago:
Yeah. Unfortunately fish stores will sell you just about anything to get your money. And they make considerably more selling cheap tanks to a bunch of people than they do selling proper tanks to the few people who have actually done their homework on tank size.
Another common myth is that small tanks are lower maintenance. If you put three drops of red food coloring in your 1g bowl, it would probably turn the water bright red. If you put 3 drops of red food coloring in a 5g tank, it would only turn the water a light pink. The same goes for the waste your fish produces. The smaller the tank, the stronger the wase will be. So you have to clean it more often. If you are looking for a truly good beginner size, I strongly recommend you grab a 5g tank.
Furthermore, there are no heaters on the market that will heat a 1g to the right temperature. Since it is such a small amount of water even the smallest of heaters will often make the water too hot.
So basically your fish is getting sick because it is being poisoned by it's own waste. Which produces a toxin called ammonia. You may recognize ammonia as a main ingredient in aggressive cleaning agents and hair bleachers. Not good lol. In order to keep ammonia at a safe level in a 1/2g tank you would have to clean it twice a day! That's probably more time than you'd spend enjoying your fish!!!
2) No heat.
They are tropical fish, silly! A water temp of 80 degrees
is ideal, give or take 2 degrees. Maybe if you live in the Sahara you will be able to get away without a heater but for the rest of the world, well... you could always crank your heat up to 27 celsius or
you could jsut get a heater. As stated above, the smallest gallonage you can safely run any heater in is 1.5g (and even heaters that low-powered are rare!). When they get too cold their metabolism will slow down, like an animal going into hibernation. And bettas don't hibernate so all it does is make them really lethargic and susceptible to disease.
3) Too much food.
Assuming you are feeding your betta, say, 3 pellets at each feeding, you are feeding him too much. Remember, your betta is cold so he will not be a calorie burning machine and furthermore he has no swimming room. So all that food just sits in his stomach and sucks the life out of him. Try feeding him only 5 pellets (or even 4) per day until you can get him an executive suite.
Hope that helps, as mentioned, we all know pet stores will feed you all sorts of B.S. in order to sell you a fish and a cruddy little tank. They know it is way easier to easier to sell you a betta bowl than a real tank because of that aforementioned misconception that a smaller tank is easier to take care of!
The good news is you found us, feel free to browse around and ask questions. And don't forget that 5g Aqueon Mini-Bows are on sale for the holidays :)
PS. Here is a good place to start: http://www.tropicalfishkeeping.com/b...are-faq-20058/