Coarseness.
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Often we've been asking for things on the nose.
Like in this formulation of the comparison test.
For all n, 0 less than equal to a sub n less that equal to b sub n, if this
series the sum of the b sub n converges, then
this series the sum of the a sub n converges.
When you formulate it this way, I'm asking for this inequality to hold on the
nose that, a sub n be between 0 and b sub n for all n.
But now, we're beginning to see, that we don't need to be quite so rigid.
Well, one thing we've seen is that convergence
depends only on the tail of the series.
So, something like this, this assumption here for all n, 0 is less than
or equal to a sub n is less than or equal to b sub n.
This can be weakened.
I can get away with just assuming this, that for all large values
of little n, a sub n is between 0 and b sub n.
And I can replace the statement that, a sub n is less
than b sub n, with something with the limit of the ratios.
I'll still make the assumption that the terms are
non-negative, but instead of this inequality, I could rewrite
that to just say that, a sub n over b sub n, this ratio is between zero and one.
And the point I'm thinking about this ratio, or I could just get rid of this and
instead of thinking about this ratio being between zero and one, I could
just ask that the limit of this ratio be some finite value L.
And at that point,
[LAUGH]
I've written down one direction of the limit comparison test.
So, we've replaced the comparison test, which is really
a pretty rigid statement, with something much more flexible.
So, yeah, the limit comparison test is a
coarser version of just the regular old comparison test.
[LAUGH]
And by coarser, I actually am claiming, this is a sort of philosophical
claim, that the limit comparison test is actually better than the Comparison Test.
And it's better for two reasons.
Well, first of all, the limit comparison test is
better than just the comparison test in practice, all right?
Beyond any sort of theoretical advantage, I mean the
limit comparison test just ends up being easier to apply.
It's easier to verify this condition, than it
is to cook up some sequence of b sub n's which satisfy
this and the sum of the b sub n's converges, say, right?
This is much harder to satisfy than just some limit statement like this.
Well, let's see and expose an example in
practice where, where this really is much easier.
So, here's an example of a series. Let's analyze the convergence of this.
The sum n goes from 1 to infinity, of this fraction
n squared plus the square root of n minus n divided
by n to the 4th plus n squared minus n. Does this series converge or diverge?
What I can do is build a sequence out of this.
So, I'm going to call the terms here, a sub n, and then I'm going to
look at this other sequence, sequence b sub n equals 1 over n squared.
And I'm going to compute, the limit of
the ratio between the terms in these sequences.
So, what's the limit as n goes to infinity,
of a sub n over b sub n? Well, dividing by 1 over n, is the same
as just multiplying by the reciprocal. So, I'm really asking about this limit.
But this isn't so painful now, right?
because now, I'm multiplying n squared times this.
So, this limit is this fractions limit, this fractions, n to the
4th plus lower order terms n squared times n squared plus lower order
things, and the denominator is the same, it's just n to the
4th plus lower order of things, in this case n squared minus n.
So, what's this limit?
Well I have got, n to the 4th plus lower order terms divided by
n to the 4th plus lower order terms, this limit ends up being 1.
So, the limit comparison test applies and
[LAUGH]
what I know, is that this series, the sum of the b sub n's
converges because this is just the p series with p equals 2.
And that means as a result of the limit comparison test,
I can then conclude that this very complicated series also converges.
But it's not just that it's easier to apply, right?
There's a second reason, why this kind of
coarse thinking is really beneficial?
And the reason is that thinking this way actually
gets us closer to understanding what convergence really means.
Alright? What's conversions all about?
Well, it's certainly not about the first million series, right?
Convergence only depends on the tail.
And what convergence really depends on is,
how quickly the terms are decaying to zero.
So, something like the limit comparison test, that asks you to look at the limit
of the ratio of the terms in the series, is really asking
you to think about how quickly those terms are decaying to zero.
And that's good, right?
That means the limit comparison test isn't just a way to get the answers more easily.
The limit comparison test is, is really
showing us something about what convergence really means.
And that's the goal of mathematics.
The goal of mathematics isn't just answers.
It's understanding.
Well, think back to that example again, alright?
This example wasn't just answering a question.
This example is demonstrating this philosophy, right?
That, I'm doing more than just answering the question of this series convergence.
I'm saying that there's a reason, an
understandable reason why this series should converge.
This series converges because it resembles, in the sense of
this limit between the terms in these series, but this series
resembles 1 over n squared.
And this p series, where p equals 2 converges,
it's where the limit comparison test this series does too.
But it's not just a calculation.
That's not just an answer. That's a reason,
why something that looks
like this ought to converge
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