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].