Welcome back.
So where we left off in the last video, I'd shown you
this thing called the geometric series.
And, you know, we could have some base a.
It could be any number.
It could be 1/2, it could be 10.
But that's just-- but some number.
And we keep taking it to increasing exponents, and we
sum them up, and this is called a geometric series.
And so I want to figure out the sum of a geometric series of,
you know, when I have some base a, and I go up to some
number a to the n.
What-- is this a to the-- why did I write
a to n minus 2 there?
That should be a to the big N.
My brain must have been malfunctioning in
the previous video.
That always happens when I start running out of time.
But anyway.
Let's go back to this.
So I defined s as this geometric sum.
Now I'm going to define another sum.
And that sum I'm going to define as a times s.
And that equals-- well, that's just going to be a times
this exact sum, right?
And that's the same a as this a, right?
That a is the same as this a.
So what's a times this whole thing?
Well, it's the a times a to the zero is-- let me
write it down for you.
So this'll be a because I just distribute the a, right? a
times a to the zero, plus a times a to the 1, plus a times
a squared, plus all the way a times a to the n minus one,
plus a times a to the n.
I just took an a and I distributed it along
this whole sum.
But what is this equal to?
Well, this is equal to a times a to the zero.
That's a one-- a to the first power-- plus a squared, plus a
cubed, plus a to the n, right?
Because you just add the exponents, a to the n.
Plus a to the n plus 1.
So this is as.
And we saw before that s is just our original sum.
That is just a to the zero, plus a to the 1, plus a
squared, plus up, up, up, up.
All the way to plus a to the n, right?
So let me ask you a question.
What happens if I subtract this from that?
What happens?
If I say, as minus s.
Well, I subtracted this from here, on the left hand side.
What happens on the right hand side?
Well, all of these become negative, right?
Let me do it in a bold color.
This becomes-- because I'm subtracting-- negative,
negative, these are all negatives.
Negative.
Well, a to the first, minus a to the first.
That crosses out. a squared minus a squared crosses
out. a to the third, it'll all cross out.
All the way up to a to the n, right?
So what are we left with?
We're just left with minus a to the zero, right?
We're just left with that term.
And we're just left with that term.
Plus a to the n plus 1.
And of course, what's a to the zero?
That's just 1.
So we have a times s minus s is equal to a to
the n plus 1 minus 1.
And now let's distribute the s out.
So we get s times a minus 1 is equal to a to the n
plus 1 minus 1, right?
And then what do we get?
Well, we can just divide both sides by a minus 1.
Let me erase some of this stuff on top.
I think I can safely erase all of this, really.
Well, I don't want to erase that much.
I want to erase this stuff.
That's good enough.
OK.
So I have just-- dividing both sides of this equation by a
minus 1, I get s is equal to a to the n plus 1 minus
1 over a minus 1.
So where did that get us?
We defined the geometric series as equal to the sum.
From k is equal to 0, to n of a to the k.
And now we've just derived a formula for what that
sum ends up being.
Equals a to the n plus 1 minus 1 over a minus 1.
And why is this useful?
We now know, if I were to say, well, what is-- let me clean
up all of this, as well.
Let me clean up all of this and we can-- OK.
So if I said, you figure out the sum of, I don't know, the
powers of 3 up to 3 to the, I don't know, 3 to
the tenth power.
So, you know, 3.
So 3 to the zero, plus 3 to the one, plus 3 squared, plus all
the way to 3 to the tenth.
So this is the same thing as the sum of k equals zero
to 10, of 3 to the k.
Right?
So this formula we just figured out, a is 3 and n is 10.
So this sum is just going to be equal to 3 to the eleventh
power minus 1 over 3 minus 1.
Which equals-- well, I don't know what 3 to
the eleventh power is.
Minus 1 over 2.
So that's kind of useful.
That is a number.
Although you'd have to memorize your exponent tables to the
eleventh power to do that.
But I think you get the idea.
This is especially useful if we were dealing with-- well, if
the base was a power of ten, it would be very, very easy.
But what I actually want to do now is I want to take this and
say, well, what happens if n goes to infinity?
Let me show you.
So what happens?
So there's two types of series that we can take-- that's
not what I wanted to do.
There are two types of series that we can take that we
can find the sums of.
There's finite series, and infinite series.
And in order for an infinite series to come up to a sum
that's not infinity, they need to-- what we say--
they need to converge.
And if you think about what has to happen for them to converge,
every next digit has to essentially get smaller and
smaller and smaller, as we go towards infinity.
So let's say that a is a fraction.
a is 1/2.
So how does a geometric series look like if we have 1/2 there?
So let's say that we're taking the geometric series from k
is equal to 0 to infinity.
So this is neat.
We're going to take an infinite sum, an infinite number of
terms, and let's see if we can actually get an actual number.
You know, we take an infinite thing, add it up, and it
actually adds up to a finite thing.
This has always amazed me.
And the base now is going to be 1/2.
It's 1/2 and it's going to be 1/2 to the k power.
So this is going to be what?
1/2 to the zero, plus 1/2, plus-- what's 1/2 squared?
Plus 1/4, plus 1/8, plus 1/16.
So as you see, each term is getting a lot, lot smaller.
It's getting half of the previous term.
Well, let's say, what happens if this wasn't infinity?
What happens if this was n?
Well, then we'd get plus 1 over 2 to the n, right?
1/2 to the n is the same thing as 1 over 2 to the n.
And if we look at the formula we figured out, we would say,
well, that is just equal to 1/2 to the n plus 1, minus
1, over 1/2 minus one.
And that would be our answer.
We'd have to know what n is.
But now we want to know what happens if we go to infinity.
So this is essentially a limit problem.
What happens-- what's the limit, as n goes to infinity,
of 1/2 to the n plus one minus 1 over 1/2 minus 1?
Well, all of these are constant terms, so nothing happens.
So what happens as this term, right here, goes to infinity?
What's 1/2 to the infinity power?
Well, that's zero.
That's an unbelievably small number.
Take 1/2 to arbitrarily large exponents, this just goes to 0.
And so what are we left with?
We're just left with this equals minus 1 over 1/2 minus
1, or we could multiply the top and the bottom by negative 1.
And we get 1 over 1 minus 1/2.
Which equals 1 over 1/2, which is equal to 2.
I find that amazing.
If I add 0 plus 1/2 plus 1/4 plus 1/8 plus 1/16 and I never
stop-- I go to infinity-- and not infinity, but I go to 1
over essentially 2 to the infinity-- I end up with
this neat and clean number.
2.
And this might be a little project for you, to actually
draw it out into like maybe a pie and see what happens as
you keep adding smaller and smaller pieces to the pie.
But it never ceases to amaze me, that I added an infinite
number of terms, right?
This was infinity.
And I got a finite number.
I got a finite number.
Anyway, we ran out of time.
See you soon.