For this question we want to determine
which of the following alternating
series converge and which ones converge
absolutely. We've got one here already a
and we'll move on to the other ones
afterwards. We're already told that this
series is alternating but you can see if
you wanted to check this yourself. It's a
-1^k multiplied by a
positive function in 1/lnk. So
this is an alternating series we'd like
to know whether or not it converges.
The question also asked us to determine
whether it converges absolutely and if
it converges absolutely then it
converges so we might be best to check
that first. And even before we start that,
it's usually good to check for a series
whether or not it has any chance of
converging. We're using the nth term test.
A series remember is summing up an
infinite list of numbers and if there's
any chance of getting a finite answer
surely as as here k goes to infinity the
size of these terms must be going to
0. So let's just check that first. So
I'll just make a note of that so as k
goes to infinity and terms in this
series do in fact go to 0.
So the nth term test isn't going to tell
us anything it's not telling us it
diverges.
And then the nth term test not telling us
anything is a good thing. You can only
tell us that it diverges if this didn't
go to 0. Okay so we're going to check
the absolute convergence first because
that may save us time. If the series
turns out to be absolutely convergent
then it's convergent. Okay so we'll do
that first and absolutely convergent
means that the sum with these absolute
value of these terms converges. So let's
check that and first of all
just know the one thing about the size
of those terms so if we take the
absolute value of those terms its
absolute value of -1^k/lnk
that's equal to 1/lnk and the
log of K is always smaller than k, so
1/lnk is larger than 1/k and 1/k
is a positive number so what we
have here if we were to take the
absolute value terms in the series we
would have a series of terms which were
all bigger than 1/k and we know
that summing 1/k in the series is
divergent. That's the harmonic series.
It's a well-known divergent series and
these terms are larger than that so they
will diverge as well so this
inequality together with it's a
well-known fact that that the sum of 1/k.
This so that the
comparison test tells us that the sum or
the absolute value of these terms
converge diverges so so my comparison
test sum of our 1/lnk diverges.
And hence your series we're interested
in.
So we've seen that this series is not
absolutely convergent. We'd like to test
whether it's convergent and for that
because it's an alternating series we'll
apply a Leibniz test.
The first thing you would check is it
alternating. We're told that you can see
that in the form here -1^k times
something positive. So let's just make a
note of these things. It's alternating.
The next thing we need to check is that
the terms in the series are not
increasing and so the
terms the series -1^k * 1/lnk
and I'll just say this is
not increasing. Okay I think for a
function of a simple function like this
it's fine just to state that it's
not increasing. It's lnk is increasing
1 over it is not increasing. Okay so so
in fact decreasing a little tick. And the
third thing we need is that not only the
terms decreasing or not increasing that
they go to 0 as k goes to infinity.
So let's just make a note of that. lnk goes
to 0 as k goes to infinity. And so all of
these three things together tell us that
the series is convergent by Leibniz' test.
Ok so,
so here we have another alternating
series and again we'd like to check
whether it converges and whether it
converges absolutely and as before the
first thing to do is to check whether or
not the terms and the series are going
to 0. In other words use the nth term
test to see whether or not the series
has got any chance of convergence. So
let's first of all give the terms a name.
So all right the series is -1^k*ak
where ak is k^k/k+1^k.
Maybe it's useful to
write it like k/k+1^k.
Now this series we really want to know
does it have a limit of 0 and so let's
look for its limit.
And this is a sort of limit that you
need a little bit of work on if you take
the log of this expression here you'll bring
the k down the front or then you can use
L'hopital's rule to find out the find
out the limit. I'll just tell you this is
a limit that you may have seen before. It
turns out to be 1/e. The important
thing about 1/e is it's not 0. It's
not 0 then these terms here are not
going to 0 as k goes to infinity and so
there's no chance that that this series
or in fact I've only check the absolute
value of this it's not certainly no
chance that it's absolutely convergent.
And also if I put a -1^k in
front of this it's still not going to. It's
not going to have the same limit but it's
certainly not going to have a limit of 0. So
by the nth term test we can see that
this series is neither convergent nor
absolutely convergent.
