### Video Transcript

Is the alternating harmonic series
the sum from π equals one to β of negative one to the power of π add one
multiplied by one over π absolutely convergent, conditionally convergent, or
divergent?

Letβs firstly remember that a
series is absolutely convergent if the series of absolute values is convergent. And a series is conditionally
convergent if the series of absolute values diverges but the series converges. And otherwise, the series is
divergent. So letβs start by testing for
absolute convergence. We can see that negative one raised
to the power of π add one is always going to give us one when we take the absolute
value. So this is in fact the same as the
sum from π equals one to β of one over π. But this is actually a series that
weβre familiar with. Itβs the harmonic series. And we know that the harmonic
series diverges. So the alternating harmonic series
is not absolutely convergent. But is it conditionally convergent
or divergent?

So our next step is to test the
alternating harmonic series for convergence. Because thatβs an alternating
series, we can do this with the alternating series test. Recall this says that for an
alternating series, the sum of negative one to the power of π add one multiplied by
π π if π π is decreasing and the limit as π approaches β of π π is equal to
zero, then π π is convergent. So for the alternating harmonic
series, we can say that π π equals one over π. So is π π decreasing? Well, as π increases, one over π
does decrease. So that condition is satisfied. But does the limit as π approaches
β of π π equal zero? Well, the limit as π approaches β
of one over π is going to be one over β, which we know is zero. So that condition is satisfied. So because we found that the
alternating harmonic series is not absolutely convergent, but it is convergent, we
can conclude that the alternating harmonic series is conditionally convergent.

We can summarize the check for
absolute convergence, conditional convergence, and divergence in a helpful
diagram. Letβs say we want to find out
whether the series π π is absolutely convergent, conditionally convergent, or
divergent. We begin by testing whether the
series of absolute values is convergent or divergent. Letβs say that we find that the
series of absolute values is convergent. Then, the series π π is
absolutely convergent. But if we find that the series of
absolute values is divergent, then the series π π is not absolutely
convergent. But it may still be conditionally
convergent. So we try a different test on the
series π π to check for convergence, for example, the alternating series test.

And if we find that the series π
π converges, then we say that the series π π is conditionally convergent. But if we find that the series π
π diverges, then we conclude that the series π π is divergent. So these are the three possible
conclusions that we can draw.