In this video, I will introduce the formal definition of series.
In English, the word "series" and the word "sequence"
are used pretty much interchangeably,
but they mean something different when we use them in math.
When we say a series, we mean an infinite sum,
whereas when we say a sequence, we mean an infinite list.
It's a simple difference perhaps.
Here is what an infinite sum may look like.
I have these numbers a₁, a₂, a₃, and so on, and I want to add them up;
and here are the same numbers just as a list.
They could be the same numbers,
but when I think of a sequence, I'm not trying to add them.
I probably want to compute the limit of this list.
And when I write them as a series,
I want to add them and I don't want to compute the limit.
I want to compute the value of the sum.
Of course the problem is that a priori,
I don't know what it means to add up infinitely many numbers.
I know what it means to add 2, or 3, or 10.
That's what I'm going to define in this video:
what it means to add up infinitely many numbers,
and I will motivate the definition with an example.
Let's look at this sum: the sum from n = 1 to ∞, of 1/2ⁿ.
The first few terms for this sum are:
when n = 1, it's 1/2, + when n = 2, it's 1/4,
+ when n = 3, it's 1/8, and then 1/16, and so on, and I never stop.
I'm going to try to add this sum geometrically.
I have drawn a square here,
and I'm going to pretend that the area of this square is 1.
And to the right, I have a second square with the same area;
and I'm going to add pieces in this second square with area 1/2, 1/4, 1/8,
and see which region I end up filling.
First, I'm going to add 1/2 of the square,
then I'm going to add 1/4 of the square.
Next, I have to add 1/8 of the square.
1/8 could be half of what is left so it could be this piece.
And then I have to add 1/16 of the square, and I'll keep going...
If I continue doing that, one thing I notice is that no matter where I stop,
I will always stay within this square.
It's just the part that I have left will be smaller and smaller.
And in a way, if I continue adding on forever,
I will end up filling the full square,
so perhaps the value of this sum should be 1.
But what am I doing here exactly?
I never really added ALL the terms (infinitely many of them),
I just added a few. I added one, then two, then three, then four,
and I looked for a pattern.
I saw what that was tending towards.
What I actually did was construct a sequence
by adding the first one, the first two, the first three, the first four,
and then trying to compute the limit of that sequence.
Let me make that more precise.
Here is the same sum again, and I'm going to add these terms one at a time,
again. But before I did it geometrically, and now I'm going to do it algebraically.
If I call S the sum of all of them,
I'm going to call S₁ the sum of simply the first, which is just 1/2.
Next I'm going to add the first two, and I'm going to call that S₂,
and that is 1/2 + 1/4, which comes out to be 3/4.
Then I'm going to add the first three. I will call that S₃.
That is 1/2 + 1/4 + 1/8, and that is 7/8.
And if I continue doing that,
then in general what I'm going to do is add the first k terms,
and I'm going to call that Sₖ.
So there could be 1/2 + 1/4, and if we put k terms,
I end up with 1/2ᵏ.
Or if you prefer, with sigma notation,
this is the sum from n = 1 to k, of 1/2ⁿ.
And then, how do we define the sum of infinitely many terms?
On the full sum S, it's going to be the sum of the first k terms,
and then take the limit as k → ∞.
If the pattern of the first three sums continues,
the sum of the first k terms appears to be 1 - 1/2ᵏ;
(That works for 1/2, 3/4, and 7/8.)
and therefore, when I now take the limit as k → ∞ here, I will get 1,
as I expected.
And this is exactly the idea we're going to take to define in general.
To add up infinitely many terms,
I simply add up the first k terms, and then I take the limit when k → ∞.
Here is the definition in general.
I want to add up the series (the infinite sum) from n = 1 to ∞, of aₙ.
First, I'm going to construct an auxiliary sequence:
the sequence of partial sums.
I'm going to call it Sₖ, where k goes to the same values.
S₁ is simply the first term in the series,
S₂ is the sum of the first two terms in the series,
S₃ is the sum of the first three terms in the series, and so on...
Sₖ is the sum of the first k terms in the series.
And then I look at this now sequence (S₁, S₂, S₃ is a SEQUENCE, not a series),
and I compute its limit.
And I'm going to say that this sum from n = 1 to ∞, of aₙ,
is the limit as k → ∞, of Sₖ.
Now the first part, computing finite sums, is no problem.
That's something we can always do.
But the second part, the limit, may or may not exist, like all limits.
If this limit exists, then we say the series is CONVERGENT,
and that means in that case, the series represents a number.
But if the limit does not exist we say the series is DIVERGENT,
and that means in that case, the series is not a number.
Now the limit may not exist for all the reasons that limits may not exist.
Perhaps it is ∞, perhaps it is -∞, or something else.
(perhaps it oscillates in some way)
In all of those cases, the series is not convergent.
In all of those cases, we cannot pretend that the series represents a number.
In the next few videos,
I will present various examples of convergent and divergent series,
computed directly from the definition.
Finally, to conclude this video, if you are familiar with improper integrals,
you may be having a sense of deja vu; and that is entirely justified,
because improper integrals and series are very similar objects.
An improper integral may have this form:
the integral of a function f, from a finite number a, to ∞.
And the problem with this integral is that it goes all the way to ∞;
I know how to integrate between two numbers, but not all the way to ∞.
Whereas the series may be an infinite sum like this one,
starting at 1, going all the way to ∞.
Again, the problem is that I know how to add up finitely many terms,
but not infinitely many terms;
and we solve both problems the same way.
The way I define an improper integral like this one,
is by first computing a regular proper integral from a to b,
and then taking the limit as b → ∞.
So I define an improper integral as a limit of proper integrals,
because proper integrals are the ones I know how to compute,
under the limit that may or may not exist.
Similarly, the way I define an infinite sum, is as a finite sum,
then I take the limit as the number of terms I add → ∞.
So I define an infinite sum as a limit of finite sums,
because finite sums are the ones I know how to compute.