Welcome to a video on the alternating series test for convergence.
The alternating series test states,
given an alternating series
in one of these two forms,
if the limit as n approaches infinity a sub n equal to zero,
Notice, a sub n, is a non-alternaing part the formula
that generates terms in the series.
And, if a sub n plus one is less than or equal to a sub n,
meaning the series is decreasing,
than alternating series does converge.
So, if these two conditions are met,
for alternating series,
the series converges.
Now, you do have to be careful about interpreting this test.
It is only for alternating series
we cannot apply it to,
non-alternating series.
Let's go and take a look at some examples,
first thing we should notices that
have an alternating series because we have
native one rise the power of n plus one
This should also remind you of a harmonic series
except now, this is the alternating harmonic series.
Here the two conditions that must be met in order to satisfy the alternating series test.
Let's start by determining the limit,
as n approaches infinity of a sub n
Again, a sub n with the non-alternating part of this formula,
so a sub n will be one over n.
So, the first condition as this limit must equal zero,
and it does.
next we need to make sure that a sub n plus one is less than or equal to a sub n
A sub n plus one would be, one over n plus one,
and that must be less or equal to
a sub n, which is just one over n
Looking at this, we can tell it this fraction has
larger denominators and therefore would always be,
less than or equal to,
one over and so these two conditions are met.
So. by the alternating series test
the given serious,
converges.
This is kind of interesting, because the harmonic series diverges,
but the alternating harmonic series converges based upon this test
Let's take a look at another alternating series
Again, our first step is to identify a sub n,
which is a non-alternating part of this formula
so a sub n,
is going to equal two n
divided by five n minus thee,
this limit has to be equal to zero, before we can proceed,
with the alternating series test.
and it's not,
notice, the degree of the numerator denominator are the same,
so this limit is equal to the ratio of the leading coefficient
which is two-fifths.
so this doesn't equal zero. So, it fails the alternating series test
so not only this limit not equal to zero,
it fails the nth term divergent test.
So, this alternating series diverges.
Lets take a look at one more.
So, first will check to see to make sure this limit is equal to zero
and then we'll check to see if the non alternating series is decreasing.
So, a sub n,
is going to equal
one over three times natural log n
Again, remember a sub n is non-alternating part of this formula.
now will determine the limit as n approaches infinity,
now, denominator increases without bound,
and our numerator is fixed. This is equal to zero
So we've satisfied the first condition for the alternating series test
and now we need to check to make sure that a sub n plus one
is less or equal to a sub n.
well, if this is a sub n, then a sub n plus one would be one over three
times
the natural log of
n plus one,
I wanna make sure this is less than or equal to
a sub n, which is one over three,
times natural n.
And again, if we're adding one to n, and then taking the natural log
the denominators will be larger,
than denominators here.
Therefore, these fractions were always be
less than or equal to,
these fractions here.
So, both conditions are satisfied,
so we can conclude from the alternating series test,
of the given series does converge.
Hope you found this video helpful, thank you for watching