In this video, I will explain how improper integrals of positive functions
are a little bit easier to study than improper integrals in general,
and I will introduce a tool, the basic comparison test,
which allows us, in some cases,
to determine whether an improper integral is convergent or not relatively quickly.
Let's recall what we know about improper integrals in general.
I take a function f that's continuous on an interval
from a real number, a, to infinity,
and I'm going to look at the improper integral
which is the integral from a to infinity of this function.
And as you know, we define this as a limit:
the limit as b approaches infinity of the integral from a to b of the same function.
That's because we can always find the integral from a to b of the function,
since it's continuous.
This limit, like all limits, may or may not exist,
and we say that the improper integral is convergent when the limit exists,
when it is a number.
That's the only case in which this integral is a number.
And we say it is divergent otherwise, when the limit does not exist.
But if the limit doesn't exist, it may not exist for different reasons.
Perhaps it is divergent to infinity, or perhaps to minus infinity,
or something else, which I'm going to call oscillating.
I put this in quotes because it's not a technical term; I haven't defined it.
Let's agree to say oscillating means divergent in any way
that is not to infinity and not to minus infinity.
So anything else.
For example, the integral of sin(x) from 0 to infinity
would be oscillating in this way.
That's the general case.
Now, how do things change if the function is positive?
Let's assume f is always positive.
In this case, out of the four cases, only two are possible.
The improper integral now may be convergent (still a number)
or divergent to infinity,
and that's it. The other two cases can't happen.
If you think geometrically, think of the integral as area under the curve.
It can't possibly be minus infinity,
because for that we would need to have negative area,
we would need to have the function be negative.
And it can also not be oscillating, because that would require
having regions with positive area and regions with negative area.
So those are the only two cases.
Geometrically, it makes sense.
We can also prove it.
The proof could be something like this.
I'm going to give a name to the integral from a to b, where b is a number.
I'm going to call this F(b). This is a function of b.
And then the improper integral is defined as
the limit as b approaches infinity of this new function F(b).
Now, what we know from the fundamental theorem of calculus,
is that the derivative of F is f.
And we also know that f is positive; that's the assumption.
That means that F is increasing.
So I'm looking for the limit as b approaches infinity of F(b),
and I know that big F is increasing.
This tells me that the only options are for the limit to exist or to be infinity.
More precisely, if I assume the function F is bounded above (picture it),
it's a function that is increasing and bounded above.
So as b approaches infinity, there must be a limit,
and that limit actually will be the supremum of the function.
This is the Monotone Convergence Theorem.
Perhaps you are familiar with the Monotone Convergence Theorem for sequences,
it's most commonly stated,
but the very same theorem works in exactly the same way
for limits of functions at infinity, and in fact is the same proof.
Or the other case is when the function is not bounded above.
Again picture it: IF the function is increasing and not bounded above,
THEN as the variable approaches infinity, the limit must be infinity,
so it doesn't exist because it's infinity.
And again these two are theorems that we more commonly associate with sequences,
but the very same thing works for limits of function at infinity,
and the proof is exactly the same.
So that's it. That's how we prove that these are the only two cases.
There is some notation that we use due to this simplification.
And very important:
this notation only applies for improper integrals of positive functions.
Never use it in any other case.
It only applies for improper integrals of positive functions.
If the function is positive,
we're going to say that every improper integral
is either infinity or less than infinity.
Infinity will mean that it is divergent,
and less than infinity will mean that it is convergent.
But only in this case!
In any other case,
saying that it's not infinity does not guarantee convergence.
But in this case, saying that it's not infinity guarantees convergence.
I like this notation because it is simple,
and because it easily summarizes the fact that these are the only two options.
When is this useful?
In some cases,
we may not need to know what the exact value of an improper integral is,
and all we care about is knowing whether it is convergent,
meaning it is a well defined number, or not.
And in that case, if I only need to know whether it's convergent or divergent
and I am working with positive functions,
there is a very useful tool
which is very much summarized or inspired by this notation.
It is called the basic comparison test, and this is what the theorem says.
I have a real number, a,
and two functions f and g, continuous from a to infinity.
I'm going to assume that f is always less than or equal to g,
and they are both positive.
Actually. I'm writing here that they are non-negative.
So far I have been working with positive functions,
but everything works the same for non-negative functions,
and nothing changes, so I'm going to be careless in the distinction.
We could have said non-negative from the beginning.
Now if I'm in this situation,
keeping in mind that the improper integrals of both f and g
must be either convergent or infinity,
if I assume that the integral of f is infinity,
well, the integral of g must be greater or equal, so it also is infinity.
IF the integral of f is divergent, the one of g is divergent.
On the other hand, IF the integral of g is convergent, less than infinity,
THEN the integral of f, since it's smaller, must also be convergent.
Now, this theorem is going to be useful when I have two improper integrals:
one that I want to know whether it's convergent or divergent,
and another one that I already know whether it's convergent or divergent.
This is been quite theoretical; what I need now are some examples.
But this video is long enough so far, so I'm going to pause here,
and in the next video I will explain some examples of how to use this theorem
to quickly determine whether certain improper integrals
are convergent or divergent.