does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.
In: 248
In a general sense, there is nothing special about an infinite series, some converge on a number, some climb to infinity. This idea is known as a limit, and really only applies in the abstract, because you can’t literally do something an infinite number of times, but you can imagine that you can. If at any point you decide to stop adding stuff, it’s no longer an infinite series and instead actually does have a discrete answer.
As for your second question, no it goes not only apply to numbers less than zero, limits of complex formulas can converge on anything, including infinity itself.
To give real life context to your own example, say you start with a glass of water. Every time you take a sip, you drink half of what is left. Immediately you can see that this will become a problem in a practical sense, because eventually you’ll have a tiny drop of water, and your next sip would only drink half of that drop. No matter how many times you go back in for a sip, you don’t actually drink the rest of the water, there is always a small amount left. We both know that in real life you’ll eventually get to 0 water left, but in the strictest since of the example, that’s not what theoretically happens. Luckily though, we can describe this problem in terms of a limit, by asking ‘what is the limit of water left if you were to take an infinite number of sips where every sip you drink half of what was remaining.’ Mathematically every sip gets you a little closer and a little closer and a little closer to zero, so we say the limit is zero. If at any point you stop drinking, let’s say after 100 sips, there is a tiny microscopic spec of water still in the glass, and we could quantify that as a traditional answer instead of calling it a limit.
There is a bar joke that comes in many different forms, but uses this same principle that goes; A man goes to a bar but is afraid of getting too drunk to make it home, so he comes up with a plan to pace himself. When he first arrives, he orders a beer. After he finished the beer, he asks the bartender to only give him half of a beer. After he finishes that, he asks the bartender to give him only a quarter of a beer, then an eighth, then a sixteenth. The bartender starts getting fed up and asks the man how long he plans to keep it up, to which the man says ‘oh I can keep this up forever’. The next day the same guy comes back to the bar and orders a beer with a grin on his face. The bartender pours the man two glasses of beer and tells him, ‘you really gotta know your limits’.
(If it’s not painfully obvious, the limit of this infinite sum is 2)
If this interests you further, I recommend checking out this old [Numberphile video](https://youtu.be/w-I6XTVZXww?si=Sy51w-sLcN–TLip) about why the infinite sum of (1+2+3+4+ …) equals -1/12
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