Comparison test - Calculus, Mathematics Video Lecture

- [Voiceover] So let's
get a basic understanding
of the comparison test when
we are trying to decide
whether a series is
converging or diverging.
So let's think of two series.
So let's say that I have
this magenta series here.
It's an infinite series from n equals one
to infinity of a sub n.
We're speaking in generalities here,
and let's have another one.
That's the series b sub n
from n equals one to infinity,
and we know some things
about these series.
The first thing we know
is that all the terms
in these series are non-negative.
So a sub n and b sub n
are greater than or equal
to zero, which tells us
that these are either
going to diverge to positive infinity
or they're going to converge
to some finite value.
They're not going to
oscillate, because you're not
going to have negative values here.
You can't go to negative
infinity, because you
don't have negative values here.
Now, let's say we also know that each
of the corresponding
terms in the first series
are less than or equal
to the corresponding term
in the second series.
Less than or equal to b sub n.
Once again, this is true for all the ns
that we care about.
So n equals one, two, three,
all the way on, and on, and on.
So the comparison test
tells us that because
all the corresponding terms
of this series are less
than the corresponding
terms here, but they're greater than zero,
that if this series converges,
the one that's larger,
if this one converges, well then the one
that is smaller than
it, or I guess when we
think about it is kind
of bounded by this one,
must also converge.
I'm not doing a formal
proof here, but hopefully
that gives you a little bit of intuition.
So the comparison test tells
us if, I guess in my brain
the larger series, the one
whose corresponding terms
are at least as large as
the ones here, if this one
converges, if this one
doesn't go unbounded
towards infinity, it sums
to some finite value,
then that tells us that the
one that is in some ways
I guess you could say
smaller must also converge.
So this one right over
here must also converge.
So that must also converge.
So why is that useful?
We'll see this in future videos.
Well if you find if you're
looking you have your
a sub n and you're like
gee, I wish I could prove
that it converges, I kind
of have a gut feeling
it converges, the comparison
test tells us, well,
just find another series
that whose corresponding
terms are at least as
large as the corresponding
terms here, and if you can
prove that one converges,
then you're good with this one.
Of course it would only apply to the case
where your original
series, each of the terms
are non-negative.
Now what if you went the other way around?
What if you could prove
that the magenta series,
the smaller one, and I
guess I could kind of
put them in quotes,
this one right over here
is the smaller, I guess
each of its corresponding
terms are smaller, what if you could prove
this one diverges?
Well if this one
diverges, it's going to go
unbounded to infinity.
It's not going to go to negative infinity.
All the terms are positive.
It's not going to diverge because
it oscillates between two values.
Once again, if it's
oscillating between values
the only way you could do that is if you
had negative terms here,
so this would kind of
be unbounded towards infinity.
Well if this one is
unbounded, each of these
corresponding terms
are larger, so this one
must also be unbounded.
So let's write that down.
So the comparison test
tells us if our smaller
series diverges, if this one diverges,
then the larger one must also diverge.
So once again, if you wanted to prove that
this thing right over
here is going to diverge,
and if you have a, once
again you know that
all the b sub ns are greater
than or equal to zero
and you want to prove it diverges,
well, maybe you could try
to find another series
where each of the
corresponding terms are less
than the corresponding
terms here and you could
prove this one diverges,
then you would be all set.
We're going to start doing
that in the next few videos.