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What is a series?

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[MUSIC].

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We're trying to capture with some precise
mathematics, and intuitive idea.

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I've got a list of numbers imagining, that
I want to add up all of these numbers.

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[UNKNOWN],

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but the list goes on forever.
Well I could just start adding.

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Here is the first number plus the second
number plus the third number

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in my list, just start adding up all of
the numbers in my list.

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But, if I am really going to add up all of
the numbers, I'd never finish.

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So what am I supposed to do?
We'll use partial sums.

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What's a partial sum?

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Well, instead of trying to consider this
series,

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where I add up all of the a sub k's.
Right?

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With this infinity up here.

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I'm instead going to consider the partial
sum.

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I'm just going to add up the first n
terms, of this sequence.

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And I'll call that s sub n.
Let's see this a bit more concretely.

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If I wanted to calculate say s sub 5, the
5th partial sum.

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Well I would just add

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up, the first five terms of my series.

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I'd add up a sub 1, a sub 2, a sub 3, a
sub 4 and a sub 5.

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If I were to calculate s sub 7, the
seventh partial sum.

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Well I'd be doing the same thing, but I'd

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be adding up the first seven terms of my
series.

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This is potentially a very confusing
point.

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I've got a sub k's, and s sub n's.

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I mean I started with the sequence a sub

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k, and out of this sequence I built this
series.

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And from this series I then started

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considering another sequence, the sequence
of partial

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sums built out of this series, which

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was itself constructed from this sequence
of numbers.

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So why is this sequence of partial sums
useful?

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Well here's why.

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I want to add up all of the terms in the
series, but I'll never finish that task.

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So instead, the partial sums are telling
me just to add up a lot of the terms.

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S of 2 is just the sum of the first 2
terms, which I could compute.

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Then I could compute the sum of the first
3 terms, s sub 3.

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Then I could compute the sum of the first
4 terms.

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I could compute the sum of the first 5
terms.

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I could compute the sum of the first 6
terms.

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I could compute the sum of the first 100
terms, the sum of the first

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1000 terms, the sum of the first 1000,000
terms.

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And if I add more and more terms,
hopefully, I'm getting closer and

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closer to what would happen, if I added up
all of the terms in the series.

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The trick, then, is to take a limit.

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I'm never going to finish adding up all of
the terms,

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but I can add up lots, and lots of terms.

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And see if I'm getting close to anything
in particular.

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Here's the official definition.

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If I want to add up all of the numbers in
my list.

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All of the a sub k's.

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Then I'm going to take a limit, of the
partial sums.

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I'm going to take the limit of adding up
the first n terms in my sequence.

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There's a bit of terminology to introduce.

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If the limit of the sequence of partial
sums exists, and is equal

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to some finite number L, then we say that
the infinite series converges.

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What if the limit doesn't exist, or what
if if the limit is infinity?

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If the limit of the partial sums doesn't
exist,

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or it's infinity Then I say that the
series diverges.

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All of this is setting up the basic
question,

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that'll occupy us for the rest of this
course.

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Given a series, does it diverge, or does
it converge?

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And if it converges, what does it converge
to?

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[MUSIC]