![]() If you can say this for any size of ε you care to choose, and |a_n - L| < ε holds, then L is the limit of the sequence. What we have in this situation is that once the index of the sequence is greater than some index value, let's call it M, the distance between nth element of the sequence, a_n, and the Limit, L, is less than epsilon, ε. Recall that a sequence is an ordered list of indexed elements, eg S=a_1, a_2, a_3.a_n, and on to infinity. Recall that we can define the distance, d, between two points as |a-b|=d. M is the index of the sequence for which, once we are past it, all terms of the sequence are within ε of L. How do we know? Well, we can say the sequence has a limit if we can show that past a certain point in the sequence, the distance between the terms of the sequence, a_n, and the limit, L, will be and stay with in some arbitrarily small distance.Įpsilon, ε, is this arbitrarily small distance. What we want is have a clear understanding of what it means to say a sequence is converging. I "learned" this in Calc I, and it's only just starting to make good sense as I try to explain it :) Sal could do (has done?) a whole video explaining epsilon stuff. As long as any a_n where n > M falls within the epsilon bounds, the series will converge. If M is 20, our epsilon bounds can be very small, and will include all the points after a_20, way off the graph to the right. If M is 0, our epsilon bounds have to be far apart, but all the a's will fall inside it (for this example). The point here is that the epsilon bounds don't have to include all the points in the series, just the points greater than M, which we choose arbitrarily. Usually it's less than one, but if we estimate that the epsilon in the video was 1, we could just as easily have chosen 1.5 and included the first couple of points in the epsilon bounds. ![]() The epsilon you choose can be any number. We still want to know when our a is close enough to L to just call it L. ![]() Say our L is 2 (this might be the L in the video). Epsilon (ε, lowercase) always stands for an arbitrarily small number, usually epsilon
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