I want to know where a sequence is heading.
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Let's think about a specific sequence. Here's an example to think about.
The sequence a sub n defined by this formula.
The numerator's 6n plus 2, and the denominator's 3n plus 3.
We can list off the first few terms.
For instance, if I want to compute a sub 1, I just plug in 1 for n.
I can get 6 times 1 plus 2 divided by 3 times
1 plus 3. That numerator is 8.
That denominator is 6.
And I could re-write 8 6ths as 4 3rds in lowest terms.
And I could keep on going by plugging in 2.
I find that a sub 2 is 14 9ths, a sub 3 is 5 3rds.
A sub 4 is 26 15ths.
A sub 5 is 16 9ths. And I could keep on going.
Those fractions, really aren't providing me with much insight.
What's the eventual value of the sequence?
Where is the sequence heading?
Might get a better sense, by plugging in, 1000.
Say, what's the 1000th term? Well that's 6
times 1000 plus 2 over 3 times 1000 plus 3.
That's 6002 over 3003. Well, that's
awfully close to 2. I can go even farther out in the sequence.
A sub a million. Is what?
It's 6 times a million plus 2 divided by 3 times
a million plus 3. And that works out to be
6,000,002 divided by
3,000,003. And that's insanely close to 2.
Alright, the limit of a sub n as
n approaches infinity is 2, to express this idea.
Okay,
so what does that really mean? Well I promise that a sub n, is as close
as you want to 2. Provided that n is large enough.
For better or for worse, these things are
usually written out with fewer words, and more symbols.
Instead of saying as close as you want to
2, I'll say that a sub n is within epsilon.
Some small positive number of 2.
And how large is large enough? Well instead of saying large enough, I'll
just that n is at least as big as some index big N.
So I can say it like this.
So for every epsilon greater than 0, there is an index, big N.
So that, whenever little n is bigger than or
equal to big N, a sub n is within epsilon
of 2.
The idea here, is that this epsilon is measuring how
close you want a sub n to be to 2.
And I'm telling you that, no matter how close you want a sub n to be to 2.
If you go out far enough in the sequence,
all the terms after that are actually that close.
Instead of writing within epsilon of 2,
you'll normally see it with absolute value.
So instead of saying within epsilon of 2, I'll say that the distance between
a sub n and 2, is less than epsilon. Put it all together.
The same that the limit of a sub n, as n approaches infinity equals 2.
Means for every epsilon greater than 0, no matter how close you want
to be to the limiting value of 2, there's some index n, so that
whenever you're farther out in the sequence than big N.
Whenever little n is bigger than or equal to Big N.
Then the absolute value of a sub n minus 2, which is measuring the
distance between a sub n and 2, that absolute value is less than epsilon.
Of course, there's nothing really special about the 2 in this.
So in general, to say that the limit of a sub
n equals L, means that for every epsilon, there's a whole number
big N.
So that whenever little n is bigger than or equal to big N, the absolute value of a
sub n minus that limiting value L, is less than epsilon.
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