What is a series?
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We're trying to capture with some precise mathematics, and intuitive idea.
I've got a list of numbers imagining, that I want to add up all of these numbers.
[UNKNOWN],
but the list goes on forever. Well I could just start adding.
Here is the first number plus the second number plus the third number
in my list, just start adding up all of the numbers in my list.
But, if I am really going to add up all of the numbers, I'd never finish.
So what am I supposed to do? We'll use partial sums.
What's a partial sum?
Well, instead of trying to consider this series,
where I add up all of the a sub k's. Right?
With this infinity up here.
I'm instead going to consider the partial sum.
I'm just going to add up the first n terms, of this sequence.
And I'll call that s sub n. Let's see this a bit more concretely.
If I wanted to calculate say s sub 5, the 5th partial sum.
Well I would just add
up, the first five terms of my series.
I'd add up a sub 1, a sub 2, a sub 3, a sub 4 and a sub 5.
If I were to calculate s sub 7, the seventh partial sum.
Well I'd be doing the same thing, but I'd
be adding up the first seven terms of my series.
This is potentially a very confusing point.
I've got a sub k's, and s sub n's.
I mean I started with the sequence a sub
k, and out of this sequence I built this series.
And from this series I then started
considering another sequence, the sequence of partial
sums built out of this series, which
was itself constructed from this sequence of numbers.
So why is this sequence of partial sums useful?
Well here's why.
I want to add up all of the terms in the series, but I'll never finish that task.
So instead, the partial sums are telling me just to add up a lot of the terms.
S of 2 is just the sum of the first 2 terms, which I could compute.
Then I could compute the sum of the first 3 terms, s sub 3.
Then I could compute the sum of the first 4 terms.
I could compute the sum of the first 5 terms.
I could compute the sum of the first 6 terms.
I could compute the sum of the first 100 terms, the sum of the first
1000 terms, the sum of the first 1000,000 terms.
And if I add more and more terms, hopefully, I'm getting closer and
closer to what would happen, if I added up all of the terms in the series.
The trick, then, is to take a limit.
I'm never going to finish adding up all of the terms,
but I can add up lots, and lots of terms.
And see if I'm getting close to anything in particular.
Here's the official definition.
If I want to add up all of the numbers in my list.
All of the a sub k's.
Then I'm going to take a limit, of the partial sums.
I'm going to take the limit of adding up the first n terms in my sequence.
There's a bit of terminology to introduce.
If the limit of the sequence of partial sums exists, and is equal
to some finite number L, then we say that the infinite series converges.
What if the limit doesn't exist, or what if if the limit is infinity?
If the limit of the partial sums doesn't exist,
or it's infinity Then I say that the series diverges.
All of this is setting up the basic question,
that'll occupy us for the rest of this course.
Given a series, does it diverge, or does it converge?
And if it converges, what does it converge to?
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