### Video Transcript

Determine whether the series the
sum from π equals one to β of negative one to the πth power over π factorial is
convergent or divergent.

If we expand this sum, remembering
that raising negative one to an odd power gives us negative one but raising negative
one to an even power gives us one, we can then see that the terms in this series
alternate between negative and positive. So, we call this an alternating
series. And to determine whether an
alternating series is convergent or divergent, we have a special test called the
alternating series test.

The alternating series test says
that an alternating series the sum from π equals one to β of negative one to the
πth power multiplied by π π converges if the limit as π approaches β of π π
equals zero and the sequence π π is decreasing. Also, note that to use the
alternating series test, we must have that π π is nonnegative for all π.

So, if we start by recognizing that
we can write our series as the sum from π equals one to β of negative one to the
πth power multiplied by one over π factorial. We can see that, for this question,
π π equals one over π factorial. The first condition that we need to
check is whether the limit as π approaches β of π π equals zero. In other words, we want to find the
limit as π approaches β of one over π factorial.

Well, as π increases, π factorial
will approach β. So, the limit of one over π
factorial will be zero. So, weβve satisfied this first
condition. So, now, we need to check whether
the sequence π π is decreasing. In other words, we must have that
π π is greater than π π plus one.

So, is one over π factorial
greater than one over π plus one factorial? Well, as π factorial will always
be less than π plus one factorial for our values of π, then one over π factorial
will always be greater than one over π plus one factorial. So, the sequence π π must be
decreasing. So, as we met both of the
conditions, we can conclude that the series is convergent by the alternating series
test.