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# Proof of infinite geometric series as a limit

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In a previous video,
we derived the formula for the sum of a
finite geometric series where a is the first term
and r is our common ratio. What I want to do
in this video is now think about the sum of an
infinite geometric series. And I've always found this
mildly mind blowing because, or actually more than mildly
mind blowing, because you're taking the sum of an infinite
things but as we see, you can actually
get a finite value depending on what
your common ratio is. So there's a couple of
ways to think about it. One is, you could
say that the sum of an infinite
geometric series is just a limit of this as n
approaches infinity. So we could say,
what is the limit as n approaches infinity of
this business, of the sum from k equals zero to n of
a times r to the k. Which would be the same
thing as taking the limit as n approaches infinity
right over here. So that would be the
same thing as the limit as n approaches infinity
of all of this business. Let me just copy
and paste that so I don't have to keep
switching colors. So copy and then paste. So what's the limit as n
approaches infinity here? Let's think about
that for a second. I encourage you to
pause the video, and I'll give you one hint. Think about it for r
is greater than one, for r is equal to
one, and actually let me make it
clear-- let's think about it for the absolute
values of r is greater than one, the absolute values
of r equal to one, and then the absolute
value of r less than one. Well, I'm assuming
you've given a go at it. So if the absolute value
of r is greater than one, as this exponent explodes,
as it approaches infinity, this number is just going
to become massively, massively huge. And so the whole thing
is just going to become, or at least you could
think of the absolute value of the whole thing,
is just going to become a very, very,
very large number. If r was equal to one,
then the denominator is going to become zero. And we're going to be
dividing by that denominator, and this formula
just breaks down. But where this formula
can be helpful, and where we can get
this to actually give us a sensical result, is when
the absolute value of r is between zero and one. We've already talked
about, we're not even dealing with the
geometric, we're not even talking about a geometric
series if r is equal to zero. So let's think about the case
where the absolute value of r is greater than zero,
and it is less than one. What's going to
happen in that case? Well, the denominator is going
to make sense, right over here. And then up here,
what's going to happen? Well, if you take something
with an absolute value less than one, and you take it
to higher and higher and higher exponents, every time you
multiply it by itself, you're going to get a number
with a smaller absolute value. So this term right over
here, this entire term, is going to go to zero
as n approaches infinity. Imagine if r was 1/2. You're talking about 1/2
to the hundredth power, 1/2 to the thousandth power,
1/2 to the millionth power, 1/2 to the billionth power. That quickly approaches zero. So this goes to zero if
the absolute value of r is less than one. So this, we could argue, would
be equal to a over one minus r. So for example, if I had
the geometric series, if I had the infinite
geometric series-- let's just have a simple one. Let's say that my
first term is one, and then each successive term
I'm going to multiply by 1/3. So it's one plus 1/3
plus 1/3 squared plus 1/3 to the third plus, and I were
to just keep on going forever. This is telling us that that
sum, this infinite sum-- I have an infinite
number of terms here-- this is a pretty
fascinating concept here-- will come out to this. It's going to be my
first term, one, over one minus my common ratio. My common ratio in
this case is 1/3. One minus 1/3, which is the
same thing as one over 2/3, which is equal to 3/2, or you
could view it as one and 1/2. That's a mildly amazing thing.