The Root Test.
[MUSIC]
Here's another Conversions Test.
So here's the Root Test.
You want to analyze a series with positive
terms.
Your suppose to calculate this limit, call
it big L.
Big L is the limit as n goes to infinity
of the nth root of the nth term.
Now, if big L is less than 1 then the
series, the sum of these events converges.
If big L is bigger than 1, then the series
diverges, and if big L is
equal to 1 then the Root Test
is inconclusive, we'll have to try
something else.
You might think that there are some
reasons to really love the Root Test.
Here's an example.
Let's look at the sum angles from 1 to
infinity of 1 over n to the nth power.
I don't know whether that converges or
diverges
[LAUGH].
I am going to use against my better
judgement, the Root Test.
So what am I supposed to do?
Well, this is a series all of whose terms
are positive.
So I'll calculate big L, the limit as n
approaches infinity of the nth
root of the nth term, well this is the
limit as n approaches infinity.
What's the nth root of nth power or just
n.
So this is the limit 1 over n as n
approaches infinity, that is 0, and 0
is less than 1.
So by the Root Test
[SOUND]
the series converges.
But in that case
I could have just used a Comparison Test,
so here is the thing to notice, okay.
1 over n to the nth power is smaller than
1 over n squared.
may be the only confusing case when n is
1, in which case these are equal, but
then when n is 2, it's 1 over 2 square is
less than equal to 1 over
2 squared.
When n equals 3, this is one over 3 cubed
which is way less than 1 over 3 squared.
Well, 1 over n to the n, as I've already
pointed out, is positive, and I know that
the sum n goes from 1 to infinity of 1
over n squared converges, that's
a P series with P equals 2.
So, by, just the Comparison Test.
Right?
I've got a series, now, which is, term
wise, less than a convergent series.
So by the Comparison Test, the
sum, 1 over n to the n, n goes from 1 to
infinity, converges as well.
And that's why i'm not so impressed with
the Root Test.
Well here's what happens right?
People go out into the world and they're
given a series
problems and sometimes those series
involved something to the nth power.
And people see you with something to the
nth power, I'm going to
apply the Root Test because then I can get
rid of the nth power.
And yeah, that's true.
But you could also have applied the Ratio
Test.
And the Ratio Test is good not just when
you've
got powers but also when you've got
factorials floating around.
And that's not just to say that the Root
Test is entirely useless.
It's just, it's not as useful I think as
people make it out to
be, right?
The Ratio Test is often easier to apply,
and it's more likely to cause some useful
cancellation.
We're gonig to see in the future some more
instances of where the Root Test does come
in
handy, but for the time being, I think
your
first inclination should be to reach for
the Ratio Test.
[NOISE].