Welcome to a series of lessons
that will explain how to determine
whether an infinite series converges or diverges.
In this video, we'll be looking at
the Nth Term Divergent Test.
So, let's take a look at our first example.
We want to determine if the series converges or diverges,
and then, state the test used.
We'll start by checking to see
if the nth term approaches zero.
If it doesn't approach zero,
then the series diverges, or is divergent,
and this is the nth term divergent test.
So, the nth term divergent test
states that if the limit, as n approaches infinity,
of a sub n, doesn't equal zero,
then the series is divergent,
where a sub n is the formula used
to determine the terms in this series.
Now, one thing to notice here is that
if this limit does equal zero,
it does not mean the series is convergent.
If this limit is zero,
the series can be convergent or divergent,
so then, we'd use another test,
but if this limit doesn't equal zero,
we can conclude the series is divergent.
So, let's start by applying the nth term divergent test.
We'd have the limit
as n approaches infinity, of the quantity, n plus four,
divided by the quantity, four n minus five.
Now, there are several ways to determine this limit.
The shortcut method is to recognize that the degree
of the numerator and the denominator are the same.
They're both degree one, and therefore, this limit
will be equal to the ratio of the leading coefficients,
and since n is equal to one n,
the ratio of the leading coefficients would just be 1/4,
and since 1/4 doesn't equal zero,
we can conclude that this infinite series diverges,
or is divergent.
Now, if you're not convinced about this shortcut method
to determine this limit, another method would be to
divide the numerator and denominator
by the highest power of n in the denominator.
So, in this case, we would divide every term by n,
so let's go ahead and show that as well.
We'd have the limit, as n approaches infinity,
of n divided by n, plus four divided by n,
over four n divided by n, minus five divided by n.
Now, if we simplify this, we'd have the limit,
as n approaches infinity,
this would just be one, plus four over n,
and the denominator would just be four,
minus five divided by n.
You'll notice that four divided by n approaches zero,
as n approaches infinity, and so does five divided by n,
leaving us with the limit of just 1/4,
which are not affected by n,
and again, this doesn't equal zero.
So, our conclusion is, by the nth term divergent test,
the infinite series diverges.
Let's take a look at a second example.
Again, the question is the same, so again,
we'll start by checking to determine
if the nth term approaches zero.
So, we want to find the limit,
as n approaches infinity, of a sub n,
which is three to the power of n,
divided by seven n-squared, minus five.
Now, in this case, notice as n approaches infinity,
both the numerator and denominator approach infinity,
which is an indeterminate form,
so to help us find this limit,
we can apply L’Hospital’s Rule,
which means, this limit is equal to
the limit of the derivative of the numerator,
divided by the derivative of the denominator.
So, this would give us the limit, as n approaches infinity,
now the derivative of three raised to the power of n,
with respect to n, we can find using this formula here,
so we'd have natural log three,
times three to the power of n.
We don't have to apply the Chain Rule because
the derivative of n with respect to n would just be one.
And now, the derivative of seven n to the second,
minus five would just be 14n.
Notice in this form, again,
both the numerator and denominator approach infinity,
so now, we'll apply L’Hospital’s Rule again.
So, this would be equal to
the limit, as n approaches infinity,
of the derivative of the numerator,
divided by the derivative of the denominator.
So, natural log three is a constant,
and then, the derivative of three to the n,
is again, natural log three, times three to the n,
and the derivative of 14n is just 14.
Well, notice in this case,
finding our denominator is a constant,
with a numerator that contains n, would approach infinity,
as n approaches infinity, and therefore,
this limit is approaching positive infinity,
which does not exist,
and of course, is not equal to zero.
Therefore, by the nth term divergent test,
the given series diverges.
So just remember, after applying a test to determine
whether an infinite series converges or diverges,
it's important that we state the outcome,
as well as state the test used to determine our conclusion.
Okay, as I mentioned earlier,
this is just the first lesson of several lessons,
if whether an infinite series converges or diverges.
I hope this was helpful.