A limit is not a process that approaches anything. It doesn't get "infinitely close". It's is exactly equal. When you write lim(x->a) f(x)=b, it means that the limit is exactly b. f(x) indeed gets closer to b as x gets closer to a but the limit doesn't change. It doesn't move around. It doesn't approach anything. It *is* b. It has always been b and it will always be b. With that out of the way:
>>11728131>>11728187These guys may sound like they know what they're talking about but they don't. Don't listen to them. A field not being complete doesn't imply that lim(x->a) f(x)=b isn't always equal to b. It implies that b doesn't even exist in that field to begin with. Say you have a sequence in the rationals (Q) that looks like it's converging to pi. We don't say "the limit is pi, but since Q isn't complete, then the limit isn't actually equal to pi" because that is a meaningless statement. How can something be equal or not to pi when pi doesn't even fucking exist in the context we're talking about? Now, regarding your specific question, as someone said before, an infinite sum (or series, for short) is just short-hand for the limit of a sum of n terms as n tends to infinity. So they're exactly the same thing. In fact, for the specific series in your pic, you have a sequence of rational numbers that converges to another rational number. Yes, that implies that the series is equal to 1. It doesn't approach it. It doesn't get near. It's not a shy guy in a party working up the courage to go ask 1 for a dance. It *is* 1. It's the same thing by a different name. Just as you make no distinction between 1/3 and 2/6, you shouldn't make one between a limit and the value used to represent it. Finally, don't fucking think for one second that we can't tell this is a thinly veiled 0.999...=1 thread. I'm just replying because it seems like you legit want to understand and aren't only a baiting retard.
