1520 11.2 #3 Geometric Series https://youtu.be/DvBMYCz9M8s https://youtu.be/UiAUGF-eDjk
One type of series where we can actually come up with a formula
for the sequence of partial sums
is called a geometric series.
You might remember learning about geometric sequences
in the previous section.
Basically, it's the same idea except the terms are being added together.
Every term is being multiplied by a common ratio, which we call r,
and that means the formula for our generic nth term would be ‘a’,
which was the first term,
and then times r to the n minus 1st power,
so that would be the formula for our series for each individual term.
But the idea here is, in some cases,
we can find a formula for the nth partial sum,
for the sum of the first n terms, and based on that,
we can determine whether this series will converge or diverge.
A couple of cases to get out of the way first,
if r is equal to 1, what would this thing look like?
Well, what it would look like is just ‘a’ and then plus another ‘a’
and then plus another ‘a’ and continuing in that fashion forever.
Now, if I just added up the first n terms here,
I would have a total of n a’s all being added together.
That would be n times ‘a’ [na].
But if we take a limit as n approaches infinity for n times ‘a’,
that limit is going to be infinite.
So if r is equal to 1, this series is going to diverge.
Hopefully, that makes sense;
if you add up an infinite number of terms and they're all staying the same,
your sum is not going to approach a value.
Basically, the only way this wouldn't diverge
is if ‘a’ is equal to 0,
and most of the time, we're going to ignore that case
because it's really not a very interesting version of a geometric series.
Next up, we're going to look at the case: if r is not equal to 1.
In this case, the sum of the first n terms
would just be all of these terms added together.
So this is our nth term.
But actually, we can do something interesting algebraically here
if directly below this, I take r times all of the terms,
because if I do that, I take r times this term,
I'm going to get ‘a’ times r.
If I take r times this term, I'll get ‘a’ times r squared.
I'm going to continue in this fashion,
and this is actually going to be the second-to-last term
because that's what I would get if I took the previous term times r.
That means, if I take this times r, I'm going to get ‘a’ times r to the n.
And it turns out, since both of these sums are finite,
(there's a set number of terms), I can combine them together.
But surprisingly, I'm actually going to subtract them.
I'm going to get Sn minus r times Sn.
My goal here is to come up with a formula for just Sn.
But the interesting thing that happens is,
if we were to subtract all of these terms from these terms,
almost all of the terms are going to cancel out.
The only ones that are going to be left are the very first term
and the very last term of this one.
Since we're solving for Sn,
it hopefully makes sense to factor that out from this side of the equation.
There's also a common factor of ‘a’
that we could factor out from this side of the equation.
Solving for Sn, we can get this by itself by dividing by 1 minus r.
The reason we don't have to worry about dividing by 0
is because we're working under the assumption
that r is not equal to 1 in this case, which gives us this formula.
What we're looking at here
is a formula for the sum of the first n terms of a geometric series
as long as our common ratio is not 1.
What I'm interested in, though,
is if the infinite series is going to converge or not.
Since we have a formula for Sn,
now we're going to take the limit of this as n approaches infinity
to see what's going to happen to the sum.
One thing you'll maybe notice here is,
there's only one part of this entire formula that has an n in it,
and it's this part right here.
This is actually something we talked about in Section 11.1.
This right here is only going to converge if the absolute value of r is less than 1.
Think about it: If this is greater than 1,
then multiplying it by itself multiple times
will cause it to basically blow up to infinity.
And if it's less than negative 1, the same problem is going to happen,
except the sign will be alternating from positive to negative.
This is only going to converge if the absolute value of r is less than 1.
Any other value for r, and this thing is going to diverge,
except of course, 1, which we've already ruled out.
Basically, this sequence, which stands for the sequence of partial sums,
converges if the absolute value of r is less than 1,
and it's going to diverge if the absolute value of r is greater than or equal to 1.
We've already excluded this case,
but if r was negative 1, we would run into the same problem.
It would still diverge.
What does it converge to?
Well, if the absolute value of r is less than 1,
then this term right here is going to approach 0 as n grows.
And that means, what we're going to be left with
is just ‘a’ in the numerator divided by 1 minus r,
and that winds up being the sum of a geometric series.
If the absolute value of r is less than 1,
then this series (the geometric series) is going to have a sum,
and that sum is going to equal ‘a’ divided by 1 minus r.
If this condition is not met,
then the geometric series is going to diverge.