![]() ![]() In the technical language of mathematics, I have specified a closeness (here, I said $\epsilon = 0.000001$, but it could have been any number, so we just call it $\epsilon$) and you have found a point in the tail (here, you found $N=537$) after which all subsequent terms (all $a_n$ for $n\geq N$) are within my specified closeness of the limit (that is, $|a_n-a|0$ if $\epsilon=0$, then that would require a sequence for which the tail is eventually constant with every term equal to $a$, not very interesting). ![]() In a specific example, maybe you found that if you go out to the $537$th term, that term and all the terms after it are within $0.000001$ of the limit. In fact, YOU don't get to choose that - I get to say how close ("within $0.000001$") and then you have to go out into the tail and find a point where the entire rest of the tail is within MY SPECIFIED CLOSENESS of the limit. How far do you need to go? Well, it depends on how close to the limit you want the tail to be. The math is just saying (in technical language) what you intuitively know: that by going far enough out into the tail of the sequence, you can guarantee that EVERY TERM IN THE TAIL FROM THAT POINT ON is as close to the limit as you want. The convergence is a property of the "tail". The important thing about a convergent sequence is that the convergent behavior has nothing to do with the first few terms it doesn't have anything to do with the first hundred terms, or the first billion terms, or any given number of terms.
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