now that we've made friends with the
idea that infinite some could indeed be
finite the natural question to ask
certainty as well
bring dozen infinite series converts
in the answer to that is a little bit
disappointing because at best we can say
it depends
there are various tests that implying
that his series convergence there also
tests to complain series divergence
such as the unit test
if one of these tests works and decides
of the series one way or the other
that's all k
uh... but there also are situations
impeach
new test
really works out then
we eat may have to give him invent a new
convergence test potentially
in this presentation of going to talk
about some
fundamental convergence test for tests
for series specifically geared towards
getting through the
ratio test as fast as possible because
the ratio test
istep test that is
making
our presentation off
our series and and most after
uh... regular
computations on power series
possible
cell
uh... that's taken away as an
introduction
again reminded their is no single
criteria that decides whether a given
series convert resort i purchased
there's no secret here in the best for
all seasons
so each of the convergence tests that
were going to look at with the
applicability some but not all
situations
and of that is then certainly also why
keep series teach mathematical reasoning
even at this early stage of calculus
when most of our focus is on modeling
visualization and participation because
what we have here is we have a bunch of
implications
where we have a b_ c_ and d_ implies
conversions and and yet and she implies
bag virgins
if we can use one of those implications
great but if we can't use any of those
implications then
are still left to you
tried to figure out something else
uh... e
common mistake i have already sent that
once in the previous presentation
but just to reiterate just because the
terrorist converge to zero does not mean
that this series converges to sell
don't do that don't
you was
that false the conclusion remember that
the harmonic series
some enables one to infinity
use in fact infinitely it has an
infinite some
in fact means
ups
convergent
you know it was something else i was
just about to say now i
don't recall
what that is so
we just go ahead
in move on
fiorello first is that we're going to
look at easter computers and test
in the computer suggests is that if we
have to infinite series some enables one
to infinity alien
in sunday nichols went to infinity bien
so that
both
the sea series have uh... non-native
terrace
zero so cynical than any in assessing it
wasn't being so be a step
bigger series if you will
uh... if then the series off that some
of the bns convergence than some of the
indians converges to
and equipment to be the contra positive
states that
they're both infinite series so that's
the russ moderately anshul article be
and for all in
into some of the indictment is
then the some of the and i purchased two
so here's what we're also have a little
bit of logic
wittiest
son bien convert just implies some of
the indian
converges to conquer positive whitby
some off the indiana doesn't convert
implies that the some of the bien
doesn't convert inventors
exactly what we have to get some of the
aliens diverging means that the some of
the d_n_c_ coach so that means
we only have to prove
one of these
technically two statements here improve
going to prove
this one
so for the comparison test
other sketches actually meant literally
let's look at
sequence off indians which will be the
blue dots here
that is
a sequence like that
is what we know it's that they'd be
installed
exceed the a_t_m_s which will be
the black dots
and we know nexium indiana both greater
equivalent zero so we have this relation
that being very quickly and
anthrax
is very positive zero that works of or
sent
every
stage so the two sequences
lineup like this if we are now
interested in
series remember that this series has to
some of these
rectangle simply erect with
uh... we use one img height being
the value at one that value add to the
valued at three and so on
so the some of the bns which mister
billing rate area
that yes
aggregators
from all
these rectangles here
because all the bns american indians to
some
off the bns
include
the some of the ams
so it must be exceeding the some of the
a m speaker s
the bacteria every
black rectangle is actually contained in
a correspondent
blue and red rectangle and so
iriarte that represents the some off the
a_n_c_'s
smaller
or equal to that some off the bns which
is represented by
this bill in reading area here
we cannot squeeze in infinity area into
a fight at a rate which means that the
some of the bns converge
if the blue and gray area is indeed
finite
then the bacteria must be fine that also
again
in a formal proof
you would actually have to explicitly
invoke
the monotonic sequence the rent
we don't do that here we just state of
more intuitive level
either way this is the end of uh...
sketch for that prove
okay so to see an example of how the
comparison test works lets take a look
at the series some in equal swung to
infinity
five in cuba months relevant square
defiant for into the six size twelve
everyone to determine if the series
convert your support divergence
now one of the things we know about the
computers intense is that applies to
series
we've uh... positive born on negative
terms
intricate c_n_b_c_ reset for any also
want to get five months twelve
which is naked seven
for in equals two with the computation
is a little bit hard to get five times
eight minus twelve times for which is
forty minus forty eight which is
negative eight
and only when in equals three
we get five times twenty seven minus
twelve times nine which is a hundred
thirty-five minus a hundred nate
which he's
positive so all of the
once where a in equal street and higher
well we have
positive terms here however
that is something that for the
comparison test
actually does not matter it is not a
problem
that the first two terms are negative
in fact is because
for convergence of tissue is very
interesting what happens for large
values often
if that means any possible any cost you
can always
outlines if necessary alternatively we
could also simply say that
we're going to consider the series only
consist of the positive terms
with the first two terms spinoff insane
apply the computers attached to it which
is completely king because all the terms
are
positive in peak and then after we have
decided whether converters archive or
just if it converges weekend
at different at the first two terms at
the end
so
first quick mental note on the
comparison test if you have a couple of
negative negative terms at the beginning
as long as he told his infinitely many
negative terms that would not be a
problem
now in terms of dot lighting the
computer is intense
uh... there are
two major steps that need to be taken
first of all
we need to figure out
want to look like these terms to
yesterday and we have to verifies that
the right inequality works
in terms of figuring out what to compare
these terms too
well if we
drop all the lower lower order to
receive we've realized that this
fractions or looks like
five in cuba divided by fourteen to the
stakes
which is about five over fourteen q
in uh... dot literally so you want to
look at this stuff because
is being killed students cities in order
to richmond will be
negligible in computers and
to the higher order trips okay so that
means that's a good point to compare the
series to fight over in cute
in reno
from pc restrictive p series with
people's three convert yes
so that means we need to verify that the
troops off the series
are smaller then five over foreign keep
now there are
many ways to do this and uh...
you'll probably want to
practice working with inequalities a lot
more then just this one example here
however
in this presentation we want to get very
quickly to the ratio test so we're going
to stay with just
one example
and uh...
so after the first step is standing up
to you know
want
term q haven't walked
you want to compare this term two int
uh... you have the direction of the
equality
which you use that if you want to
compare to something that
call over just indian equality must be
secure series that your testing is
semesters that are smaller
that what you're comparing two
well then you have to summer show that
the inequality bricks
and there are several ways to do that
one of them that i think we're the only
s legit is to simply
simplify this inequality until you have
something
that yes
correctly statute as long as we
uh... make uh... work on our stance
correctly so here for example of the
inequality
we could multiply with the denominator
on the left side
and that denominators positive which
means the inequalities not affected and
sell
the first line is equivalent to check in
with her five in cuban minus twelve is
clear
is smaller than five over four inches
sir
times for into the six just twelve
and uh... well now we can just knock out
the right hand side
so this is acquitted two checking
whether fighting cube minus logan square
smaller then
five before in q transported to the six
which is fine thank you
twelve times five over fourteen q twelve
twelve over forest three times finds
fifteen
so we get the five fifteen over
in q appearances inequality
by inspection is correct because if you
have a number it's a track something
from it and that certainly is smaller
than when you have a number
and at something to that number
nell the logical presentation the
logical derivation of this inequality
actually would have to go
from the bottom to top because logically
we have to
start with something that is correct
intro that concludes that this is
correct in saying this is correct
however
because all the steps that we have here
are reversible we will forgo these
technicalities
uh... working with inequalities is p
huge topic
that you
probably want to spend more time with
however
in this presentation that is
not one of the main goals we
want to push for some restrictions
so tumko include
this example read would say that
we can conclude because
terms of the series are bounded by five
over fourteen cute
in because
the sunday nichols went to infinity five
a fourteen q which is just five quarters
times the son equals one
to infinity whatever in q which is a
p_c_ image that convert just so because
that upper bound convert its
the series summing equals one to
infinity fries in cubed
minus twelve in square divided by
fourteen to the success twelve
that series
convergence ansatz worried
with us we're going to talk about
absolute convergence and the big note
that we have here is that the computer
doesn't taste focuses on series
with no negative terms
well this kind of suppressing the it's
signs
is actually a very common step
and it goes hand-in-hand with the
following definition
uh... a series here
some people's wanted to make the indian
is called absolutely convergent effect
only if the series some enables one to
infinity
in absolute value off the absolute
values off its terms on purchase
the language
basically indicates that if a series
converted stamps the
then the series should converge
that is something that we have to prove
just because
the language is suggestive does not mean
that the implications
the proof isn't too bad
because the key to the proof is
something that
somebody has thought about for a long
time
if i take it up just a an absolute found
a few
i no longer have negative numbers
because if he and his negative them
dishonest hero
innovators positive that this summer's
too
in absolute value and so that means it
etienne
class a_t_m_ absolute values between
zero
into a m
absolute value and this is something
we we can use
the comparison test for because
we know that some of the a m converges
absolutely which means
that there's some off the a m
absolute value convergence which means
that the some
off to a time stay an absolute value
converters also
if that means a comparison test implies
that the some of the a m plus an
absolute value converges because
they are
greater equal then zero endlessly
ripples in
it's convergence here is that we know
that means now we can use algebra and
realized that the signing equals one to
infinity alien
class be an accident that you minus to
some of the an absolute values that's
the some of the interesting in absolute
value minus a an absolute value which
leads to some of the a m
which therefore in st
convergent
series
okay
uh... as an example
the series somethin equals one to
infinity negative want to the end of
late-breaking cubed converges absolutely
attacking us
because the some off the absolute values
is just the sounding equals one to
infinity one over in cuba
which converges
by the pc restates
so
that again shows the accident
convergence
in class
convergence
now it would be very very nice if the
other direction were true also however
that is not the case which means that
they will be series for which we cannot
suppress the negative signs and the
standard example
the what is called the alternating
harmonic series the series
some in equals one to infinity negative
want to be in class one
divided by it
but we're going to show in this example
is that this series converges
but not absolutely
and uh... well
because that
absolute value of native want to be in
plus one definetly and it's just one
over in
in because the harmonic series died or
just remember that list of warning at
the beginning of this presentation
that's the one that you always want to
keep in the back up your mind
because the harmonic series di burgess
the series some in equals one to
infinity negative want to be a plus one
divided by ten
does not
converge
absolutely
for convergence
takes a little bit to work out so thats
let's look at conversions for the poor
diffuse that be para consecutive terms
in the partial sum stand
a partial sum that goes from in equals
one to end
even ending index to win
of these guys
will be the some in equals one taking
off
one over
to win minus one minus one over to win
so this is a summation
that starts
we've in equals one here being won over
two times my my responsible and over one
just at this point starts here was one
minus one over to which would be better
for equal stuart here within the next
time
uh... in it continues like that was
automatic negative signs up to the term
well over two accounts
native whatever two times capital in
which is exactly
what we
would get here if we planning to times
capital in because megan wanted to
entrust once nato one
uh... times one over
two times capital and so these two sons
really cover the same
terms but now here we can do some
algebra
in if i find a common denominator i get
to win minus two n minus one divided by
two and minus one
times to win
with the two in my list to win goes away
and we just end up with one over
to win minus one times to win
and i realize that that is necessary for
them to some in equals went to capitol
in
of one of her fourteen squier
i've just drop the minus one and that no
matter which makes
who'll fraction
larger
is that is necessary for them to some
enables one to infinity whatever in
squared
converges
by pc recessed
and so
the sequence of even number partial sums
is increasing can't balloting
which means that convergence
increasing in politics
in so that means he converges
by the monotonic sequence the rent
if you will
is
similarly we can actually shows that
sequence of all a number partial sums is
decreasing in about it we just go from
in equals one too
to in just one
if that is decreasing unbounded also
then that means
that that sequence convergence
also
into the difference between consecutive
even in our number partial sums it's
just a one over in that is being at work
subtracted and that means
they must have the same limit because
they're
difference goes to zero if that means
that limit will be though some off the
series about this kind of
alternating series approaching something
that we're going to talk about some more
when we discuss the
alternating series test
uh... however
that you see
separate test that isn't crucial
to talk about our series and as i said
i want to ask
ticket as quickly as possible to the
ratio test
we also need absolute convergence
because that is something that the ratio
test talks about it so it makes sense
too
explore a little bit ask you
how convergence
in absolute conversions are related to
each other
so in other initiatives
the ratio tests is that if you've got a
series quiz
non-zero shirts or do you know that
because you're up for all the kid
if you've been considered the limit in
going to infinity of a_t_m_ plus one
divided by eighty and absolute value
of the absolute values of portions of
course they kicked her s
then you have the following if this
limit of these quotient it's smaller
than one
that some converges absolutely
in particular that means a convergence
if the limit is greater than one than
the sum divergence
if the limit is equal to one dentist
series might convert or diverge and we
also say that the ratio test failed in
this case so again
this is something that we have a test
that says all right
it one thing happens we've got
convergence that nothing happens we
divergence
but there's still a thirties
we we don't quite know what to say
end that for case we'd actually
discussed very quickly if you look at
the harmonic series
israel is any of the p series
you know if the quotas of consecutive
terms for all the series is one
some p_c_b_'s convergent
in some ppc res
divert
so the most important part for the proof
is
to prove
parts one
in parts too
the case distinction between
whether the limit of the quotient that's
more than one
greater than one
in are
that
the in
uh... at sea
uh... that also these limits are also
then one off the reasons why
uh... is hard to work with
series then see if i can grab this
somewhere yet here
because now
not only do we have
the sequence
off the terms as well as the sequence of
the partial sums but now we also have
the sequence of these
quotient significantly affect me
have to think about
indicates distinction is different right
if the terms don't go to zero in the
series is diversion break here
we have something the week
dick argue around the limit of one so we
need to make sure
that we remember all these duty
in terms of proving the ratio test it's
actually not that bad for part one
we assume that the limit
it's more than one
and we take the average between that
limit employment so we take you being
one-half being the limit of the quotient
one half times at the end of the
coalitions
class one that's more than one
infor sufficiently large in remembered
amidst
act win-win things get large
for sufficiently large and we have that
zero so cynical than a unnecessary
prevent some number in times
cued to the infor in greater than zero
and that's because
eighteen plus one divided by any in
use smaller
thank you and that means eighteen plus
one this one i think your times any in
which means that it's what i think you
scrape in minus one and so on and so
wanted so
recursively you can alternately
derive
in inequality like this one
is therefore by the comparison test the
series convergence
absolutely and thoughts was that he it
for the first part of the ratio test
folk are two let's note that if the
limit is greater than one than for large
in we have the a m plus one absolute
value is greater than
in absolute value and so for some large
enough bien
we have the a_m_a_'s see greater than
zero and that means
for all the instant come beyond that
we have today is greater than c
in fact means the terms of the series
don't convert to zero and two series
diverges
unit test
this kind of reasoning is certainly
something that is also
on the board a rough going to advanced
calculus class workgroups
take center stage
check with your teacher how much off the
proofs will be
required of you we're not going to look
at how the ratio test acts
will have two examples the first one is
to determine if the series some in equal
zero to infinity
may get one to the end times in divided
by two to the and calling for just
so we take the limits of the quotient
off consecutive terms which are native
born to be in plus one times in class
one divide bike
tune into the address one defied by
native went to the in
times in divided by two to be an right
turn here acting
inspecting plus one
alright so if we simplifies the fraction
here multiply with the receptacles
denominator we get the limit
negative went to the end plus one times
info spawned from here
times to to the end which is that
denominator off the denominator
divided by two to the in-class want from
here
times negative want to be in times in
which is that
numerator
off the denominator
i think i size i fear could have kids
already because everything is inside an
absolute values of the native science
do anything
then we can cancel the to to the intake
institute at the entrance one
giving us was in the middle employs one
over in times one camp
interesting goes to infinity incomes one
imprint coastal fund so that in the case
one-half that it's more than one and
that means this series convert just by
the ratio test
that cm from his example so
after we actually have internalized the
ratio test applying it actually is the
fact that
and that also
happy is the case when we have a more
complicated series such as the series in
equal zero to infinity in factorial
divided by
again we set the limits in ghost
infiniti of the interest first term
which isn't as one factorial defiant
right eight to the address one
and we divide that by
instrum which is in factorial defiant by
eight
to the in
we simplify this incident it is and goes
to infinity of in class one factorial
from the numerator
time speech of the in which they can all
interested in a letter divided by eight
to the implants one which is economic
growth snow rare times in factorial
right here
now the cancellation of eight to the
universe is a two billion plus one
giving us one eight that's not a problem
something
that's care trip people that i've seen
it happen is
how do you cancel in class one factorial
in in factorial
well in class one factorial es the
product of the first in class wanting
pictures
so that's one faster times three times
all the way to end times and plus one
in fact over this one-time students free
all the way to in if you cancel all that
the only thing that remains is it in
plus one in the macgregor
so this is the new medicine goes to
infinity of in plus one over eight which
is infinity which is most certainly
greater than one
in so that means the series
divergence
the ratio test
quick note
intended series who haven't in fact
oregon to turn on the air
if you had in fact or in the denominator
then
computation what worked with the
reciprocal here sentry input
in that with the democrats
zero which would indicate colin versions
and so that means unless the numerator
behaves really badly and that can happen
uh... listen numerator behave really
badly
a taylor series actually we'll convert
and we will see that as we talk about
power series
reach
in the order in which i have
lines there's an old
uh... for my takes that these will be
the next presentation bob if you are
first interested
just convergence tests for series in
general if you want to get all of those
artists awake you wish to get the power
series and co
further convergence tests i'll leave
that up to the
instructional design of your coursework
your choice
right now what you want to do is you
want to
test some series for convergence and
divergent that you're going to use
comparison test
in the ratio test
