So for this to be true, n has toīe greater than 1 over epsilon. Sides of an inequality, you swap the inequality. Of both sides of an inequality, you would have that Same thing that 1 over n has to be less than epsilon Thing as 1 over n, as the absolute value of 1 over
Cubic sequences convergence plus#
Which is another way of saying, because this negativeġ to the n plus 1, this numerator just swapsĪbsolute value of it, this is always just The n plus 1 over n has to be less than epsilon, And a sub n is just thisīusiness right here, so it's another way of saying This to be less than epsilon? Well, this isĪnother way of saying that the absolute value of a sub Of a sub n minus 0, what needs to be true for Larger than that is going to be the case that That means that the value of our sequence forĪ given n is going to be within these two bounds. Our sequence and our limit is less than epsilon, That if n is greater than M, the distance between our Now, this is sayingįor any epsilon, we need to find an M such So what does that say? That says, hey, give Term in our sequence is going to be within epsilon of Index is greater than capital M, then the nth M greater than 0 such that if lowercase n, if our Going to be true if and only if for any epsilon It, and that's what I want to do in this video. It will get smaller and smaller and smaller. Oscillates between negative 1 and 1, it seems like Larger and larger, even though the numerator Our sequence explicitly- the limit of this as Limit of the sequence- and so I can write this as That for this sequence right over here that canīe defined explicitly in this way, that the This sequence- and this was in a previous video. Having said that, if you have to do this for another problem, you can solve the inequality to find suitable values of epsilon for the particular sequence. By looking at his graph, you can see that the first value was outside his criteria according to the epsilon he chose, but for convergence, we really care about what happens further along in the sequence. If he had chosen an epsilon of 1/3, his value of M would be 3, and it is still true that for values of n greater than M, the values of the sequence would be within the range defined by that epsilon. When he chose 1/2, the reciprocal was 2, so that was where he drew his M. Once he "chose" this value, he then used it as he talked through the proof. He could have chosen another value, but it is a good idea not to go too far along in the sequence, or you might wonder whether the proof holds only in the tail end of the sequence or perhaps something interesting happens in the first values.
In this case, Sal started drawing a line for his arbitrary value, and then said it looked like he had chosen a value of 1/2 for his epsilon. Well, epsilon is an arbitrary value that is chosen for the purpose of proving a limit.
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