We're going to determine if the infinite series
converge absolutely, converge conditionally, or diverge.
To review, a series, the summation of a sub n
is absolutely convergent if the summation
of the absolute value of a sub n also converges.
A series, the summation of a sub n
is conditionally convergent if it is convergent,
but the summation of the absolute value of a sub n diverges.
So going back to our example,
looking at the first series,
notice how we have the summation of one divided n,
or one divided by n to the first.
So if we apply the p-series test,
given here, notice, if p is greater than one,
the series converges.
If p is less than or equal to one,
the series diverges.
So here we have p equals one,
and therefore our series diverges.
Let's go and generate a few of the terms in the series.
Notice when n is one, we'd have one, plus
when n is two, we have 1/2, plus
when n is three, we have 1/3, plus
when n is four, we have 1/4, and so on.
You may recognize this as the harmonic series,
which even though terms are decreasing,
this is a divergent series.
So we'll say, by the p series test,
with p equals one, which is less than or equal to one,
the series diverges.
Now take a look at the second infinite series,
which is an alternating series.
Notice how for this series, when n equals one,
we would have negative one to the second
divided by one, which is one.
And for the second term,
we'd have plus,
when n is two, we'll have negative one divided by two,
or negative 1/2, so we could write minus 1/2.
And then when n is three,
we have negative one to the fourth
divided by three, positive 1/3,
and then when n is four,
we have negative one to the fifth,
divided by four, or minus 1/4, and so on.
Notice here, we have the alternating harmonic series.
So because we have an alternating series,
we'll apply the alternating series test.
One thing to notice here
is if we took the absolute value of the series,
we'd have the harmonic series given here,
which we know diverges,
which means, if this does converge,
but the alternating series test
would only be conditionally convergent
because the absolute value of this series diverges,
which again we see here for the basic harmonic series.
So notice, for the alternating series,
the none alternating part, a sub n,
would be equal to just one divided by n.
And now to review the alternating series test.
If we have a series in this form here,
where a sub n is greater than zero,
if the limit as n approaches infinity of a sub n equal zero,
and a sub n plus one is less than or equal to a sub n,
meaning the terms are non-increasing,
then the alternating series converges.
And again, to check for conditional or absolute convergence,
we'll determine if the summation
of the absolute value of a sub n converges or diverges,
which we already know diverges.
So to apply the alternating series test,
you notice how a sub n is greater than zero,
and now we'll take the limit as n approaches infinity
of a sub n, which is one divided by n,
notice how this limit does equal zero.
And now we'll check to make sure
that a sub n plus one is less than or equal to a sub n.
Well here the terms for a sub n equal one over n,
and notice how the terms are decreasing,
and therefore a sub n plus one is
less than or equal to a sub n,
and therefore the series converges
by the alternative series test,
but again because the absolute value of our series diverges,
which we showed above,
the alternative series is conditionally convergent.
So, let's go and summarize this.
We'll say, by the alternating series test,
the series converges, but since,
the summation of the absolute value of a sub n,
which equals the summation of a divided by n diverges,
which we know by the p-series test.
The alternating series is conditionally convergent.
If the series here had be convergent,
then we could say the alternating series
would be absolutely convergent,
but in the case, because the summation
of the absolute value of a sub n is divergent,
we have a conditionally convergent alternating series.
I hope you found this helpful.