Root Test - Calculus

Hi, everyone! Welcome back to integralcalc.com. Today we’re going to talk about how to use
the root test to determine whether or not a series converges or diverges. And in this
particular case, we’ve been given the series 3 to the k divided by the quantity k plus
1 raised to the k power and we’ve been asked to determine whether or not this series converges
or diverges when k starts at 1 and tends towards infinity.
So what I’ve done here is taken a screen-grab of the root test section of the table of convergence
test that I have on my website. And what it tells us is that it outlines how to use the
root test. Despite that this might seem a little complicated, the root test is pretty
simple. It just allows us to do some cool simplification when we have a specific type
of series. What we’re looking for with the root test is a couple of things. We want to
make sure that every single term in our series is raised to the power of k or n or whatever
you’re variable is. Every term in our series is raised to the power of k. That’s a dead
giveaway that we should be using the root test. Because what we do when we use the root
test is take the limit as k goes to infinity of the series. And this is what’s cool about
the root test. We take the entire series and we raise it to the power of 1/k or whatever
your variable is. Basically, what we’re doing is taking the kth root or the nth root
or whatever of the entire series and what that allows us to do is simplify these exponents
which is pretty cool so when every single term in your series is, in this case, raised
to the power of k, and you take the kth root of the entire series, it’s going to go ahead
and cancel for you, all of these exponents. So it’s going to cancel all of these exponents
here because here, for k and 1/k, you’ll end up with k times 1/k. The k’s cancel
and you’re just left with 1. So all of those exponents go away and when
we simplify now what we have left, we’re going to simplify and we’re going to get
some number. We’ll call it L. So when we get L, what the root test tells us is that
when L is less than 1, then the series converges absolutely. If L is greater than 1, then the
series diverges. And of course if that is equal to 1, then the root test is inconclusive
and we’ll have to use another test. So again, all you’re looking for is that each term
in your original series here is raised to the power of k or whatever it is that is the
variable you’re working with. So what we’re left with here is the limit
as k approaches infinity, now that we’ve canceled our exponents, we just have 3 over
k plus 1. And we can now very quickly see, if we plug in infinity for k, the denominator
here is going to get very, very, very large. And as we know, whenever we’re using these
convergence tests and we get really comfortable with this, whenever the denominator of a fraction
gets very large, that means that the entire fraction is going to tend toward zero. Because
if you go ahead and plug in 3 divided by 1 million or 1 billion in your calculator and
you’re going to see that’s an extremely small number, it’s going to approach zero.
So this whole function here will be zero, which means we’ll get L equals zero. Okay.
So now that we have L equals zero, we can go back up here to our root test screenshot
and we know that the series converges if the series is less than one. In this case, L is
less than one. It’s zero which is less than one so we can confirm that this series converges
absolutely. So we’ll go ahead and say converges absolutely by the root test. And that tells
us the convergence for this series. So that’s it. Pretty simple. I hope this
video helped you guys and I will see you in the next one. Bye!