the first presentation on convergence
tests was deliberately geared towards
getting to the ratio test as fast as
possible so that
uh... we could talk about power series
uh... this is the second presentation on
convergence tests it should not be
understood as something that says
here's all the other convergence tests
because
in fact use
not the case
but uh... rather then saying that this
is a presentation to just
talks about if you more
commonly used convergence tests
that can be used for example to figure
out whether any
tennessee restoring power series
converges
at the boundary of interval of
convergence we're it is not
clear whether at those points to series
convert to survivors
let's take a look at what we have
so
four days it was if you want to see it
that way the ratio test delete use
highly effective and that is also
something that they can see in the
presentation on our series
but there are many series for which the
ratio test
does not work
and some of the series of rice for
example at the bones of the n_ double
congrats our series and that means
we need further conversions test if we
want to know more details
power series for example or about series
that just come up from other context
uh... remember however
that no one test will do it all theories
no
single
convergent s convergence test that says
and i'll give me an artery series and
here is going to convert disappears when
it by bridges and the other thing that i
want you to remember is
that common mistake just because the
church go to zero it does not mean that
series convergence
so i remember the harmonic series
that some in equals one
to infinity of one over in
that is a sum of terms that go to zero
but some itself
and in fact that is a good reminder
because the alternating series status is
one that
dangerously close to something like that
that talks about
terms
whose unit is zero namely the
alternating series test
takes a sequence of numbers detained are
greater equivalent zero
whose name is equal to zero
isn't so that uh... sequence of numbers
is decreasing so be capers one is this
an article that b k i think for me that
would be
let me say known increasing but that's
one of those words that i have a hard
time wrapping my mind around
then the alternating series
some in equals one to infinity negative
want to the in plus one p in converges
sores alternating because we're putting
what is called
alternating signs in there
inside so that means is dangerously
close to saying you need zero gives you
convert inspect only does it
for the alternating series not for a
series of
positive terms all the way through
how and so yes series of this type of
cold alternating series how can you
prove something back that way up
if u
lineup
you are number axes vertically
then your first some and will be just
being one and that's positive
ins
will you then go from the first partial
sums of the second part will sound the
second part of some will be people on my
knee speech it was so
your second partial sum will be going
down
the third partial sum is d one-minute
speech you'll just be three so you're
going up again
the fourth partial sum
goes down again because now you've got
b_-one
my speech ups be three months before
that just keeps on talking like that in
because
the be case go to zero
the also nations gets more it's more so
it seems to me soon
that this ultimately stabilizers at some
unit and we can actually prove that
uh... in a mother mike lee
pre watertight fashion
beacause
we know that if we
look at the different at
number turning this series
antonov number two
in all of them are turning to see
tourists are in even number during this
series
in the auto burke terms positive the
even numbered term is negative
in their difference is greater equal
then zero because the next term is
hysterical than the previous took
uh... so that means the sequence of the
even partial sums partial sums that go
from one to
to win
that'll be a non decreasing sequence
because the sequence can be parsons
he won't just be too
the one i speak to
b three months before trust before
b_-five might is be six
in somewhat
serenity if i
number term
subtract from it
previous even number two rights are not
looking at b_-two in class three which
comes after
b_-two in just two
melt we have
that's the or number term here it is the
next one so at this point
it's smaller than atoms of these
differences necessary will then zero
because of that we have that the
sequence of the all-out partial sums
knowing
increasing beacause
i will have
that for the first time here will be
do you want
then i add to that
the freemen speech into that right at
the five months before and so on so i
had a bunch of native terms now
uh... if i care of them up
in in the right way in in
carrying out that such a way that always
period
and even longer term was an excellent
number term
int
tells me
that now because the
uh... difference between with the
number of some n_p_r_'s number
partial sum is just
truant just first term which is made a
point to the two in class one
class one
times be south to entrust one that's
going to be directly on zero we have
that's the
origin numbered
partial sums which are
known increasing were going downwards
into the
are very close them
even at number partial sums which are
non decreasing so that those are
going up which means
that we've got eight
known decreasing sequence
responded by above plan on increasing
sequence and so
by the monotonic sequenced theorem
the even in the auto partial sums
convert because they are
both monotonic
int boundaries
in because there's a difference of these
two sons
used to be to ruin
class one which goes to zero is in goes
to infinity
the sequences of the evening not partial
sums up the same limit
if that means the overall sequence of
the partial sums converges
and that means
that the series convergence
okay
yes challenging as the proof of people
in the series the riskiest is simple is
its application because
if i want to show that the series some
in equal stuart to infinity negative
want to be in defiant bienen com
burgess
well i can say is but of course it does
because
zero is less than or equal time
one over and then
in class one i guess
eyes should have said ellen in here let
me fix that go quick
the next term okay so
zero is less than one over aliens
radiance in particular also for
in an in class one
but whatever they learned in class one
is less than one hundred and and and and
that's because
and then in class one is greater than
aman and remember that taking the
reciprocal changes the inequality around
so
the other thing we know is that the
limit as time goes to infinity of one of
relevant
is equal to zero even though the normal
rhythm
goes very slowly to infinity often it
doesn't want to bring to the is zero
if that means that this series satisfy
saipan disease off the alternating
series test
namely
limit
off the terms is equal to zero
in the terms are
non-negative
uh... not increase it
application of the alternating series
test this theory
easy it also there is a beautiful one
trick pony because unless
you have the negative want to be in or
they don't want to be in class one
in your series you'd be hard-pressed
to apply the authentic series
uh... limit will be written test has a
bit
why do applicability
and political garrison test says the
following if we have to series with
positive terms
if they did not off the closure of the
in overby in is equal to
number c that is greater than zero
than either both series convert report
series diverge supposed series have
the same
convergence behavior again
here is where
idea of mathematical reasoning certainty
became enters
turned to discussion that we have a lot
series here because we now have another
test
we have another unit
will be taken june the twelfth terms and
it's not unusual for some it's in the
mail proportion but it's not a
washington
of consecutive terms and when that in
the arctic wilderness prison zero
as opposed to in the ratio test being
smaller then
very the series have the same
convergence behavior so again we
have to you
uh... you should were that that we train
our mind mathematically to the point
where we can keyed all these things
separate in our mind and applies and win
appropriate
but uh... other than that the proof
isn't too bad because with the limit off
the portion is a constant that is
greater than zero
than for sufficiently large in we have
the ams approximately equal to see times
being
in because only the truce
with large and determine whether a
series convert or diverge converges over
just
in because multiplication was a constant
does not change the cons the convergence
behavior both series at the same
convergence pediatric because basically
the sums up their tail idioms are just
constant multiples
of each
uh... in terms of applying the test
if we want to determine if the series
some in equals one to infinity
freudian square over five into the
fourth minus seven convergence
we could use the columbia is intestines
so in some ways then
the comparison test is a bit
off eight
center shortcut because it
allows us to avoid looking to
inequalities in many cases
if i wanted to call in contra
compare this series with anything it
would probably be something like
inscri over into the fourth
which is one over in square at the sum
of one over in square
so
if i forward the quotient of civilian
square divided by five
into the fourth minus seven time divide
by one over in square at will
if i take that limit i'd just multiply
with the reciprocal whatever in square
depicted in the gulf
freely into the fourth divided by five
into the with minus seven
expand with
one over into the highest our office
nominated or so we could spend with my
over into the fourth
we get to the middle thirty divided by
five minus seven over into the fourth
which is six
which is greater than zero
in therefore be close to some in equals
one to infinity
one over in square converges by the
p_c_b_'s test
the series some unique was going to
infinity thirty inscrutable five into
the fourth minus seven
converges find and compare isn't it
that's how we work
within that compares it to us
we're closing out breeze to feel runs
that talk about
how to is to make remainder terms
remember that went by
charters talking to you about company
uh... convergence tests
i mentioned that it's actually not
terribly common to actually be able to
find the exact value office series but
what we can always do is because we know
that the some over the tail
gets smaller as we make arsenic are in
the next some larger
yet one of the things that we can do is
we can estimate
how big the remainder is which would
then tell us
how close are partial sums are to be
overall value off the series and so
if we have a no negative decreasing
function
if int yes is any
series of so many post wanted to be a
coffin that convert just because of
the integral test
wealth in the some into equals in class
one
to infinity info indecency rippled and
the integral
from into infinity info fix the x
is that means that the difference
between the value of the series
it's capital gains partial sum
catherine's partial sound here
is this a ripple than this integrate
this into go maybe something
that became
estimate
compute
if we want to prove this result they
just go back
and think about how the integral test
worked
here's our function
terror softer
series are
in valuations and integer point since
then
the series itself
uh... is for the area of these
rectangles
underneath
function
that are always
having our their top right corner at
these
who points to or the evaluation
off the
off the sequence
ins so
that means that if i take the difference
between that total
int say the some up to three that would
mean that take the first three terms the
way and that means
that i have the term starting at four
five install one which is the some
off in going from in just one to
infinity f often
if by some from four to infinity in this
series than actually pick up this
rectangle here which means i also
pick up something that has to be
estimated by the integral starting at
three which is why the summit in naples
and just want to infinity of often
exactly ballot by the integral
of if a fix
starting act
in so that is
the rough idea behind
estimate
we can use that now to compute
approximation for example
for the summer off the pcbs some in
equals one to infinity one over into the
fifth
uh... within iraq is sent into the
negative four well
i would
wants to make sure that my son in equals
capital in plus one to infinity one over
into the fifth is hysterical than the
sum in court
adding the integral into infinity
whatever x to the fifth
uh... in of that is
articulate it being true to what extent
a two five that's negative
one fourth exterminated fluorinated one
over
forex to the fourth the integral will be
the into derivative at infiniti and at
in one at infinity it's zero at in and
get one of a foreign to the four th
into now i'm just going to make these
upper band listener equal to
tend to the fourth which means one of
our ford into the four th should be
necessary book
one overt into the fourth
which is the same as for into the fourth
being
greater april that into the fourth
because
when we go to
and reciprocal speed inequality
has to be changed around
and that would mean that into the floors
or acted before through first so forth
rudolph four times in has to be gray
ripples in tin and the fourth of the
forty s of that square root of two or so
in this very positive over spirit of to
attend a whisper of too
is five times route to a brute who was
one point for something
uh... one planned for something times
five is a certain point something which
means in great reporting eight
will do it
insoles and the approximation would be
the some enables one to eight
one over into the fifth which is
something that can be worked out with a
calculator
it's approximately one point zero three
six eight eight
int that means here is that
these digits are completely accurate
this one if you're around that would
actually be a nine
depending on how far we have suppressed
the error
it could also be a native
it doesn't change too much
in fact
a few featured here
so
that is an eight
pretty good approximation
for the value office p series if you
need four-digit you definitely have
enough in
that leaves us with one more
waitress may come and our terms of the
actually goes back to you alternating
series maybe
if i have to us
uh... a_t_m_ spm on increasing sequence
of non-negative numbers
internet is a series some enables one to
infinity made a point to the invoice one
a m converts because of the alternating
series tips
that some in equals in class one to
infinity
native want to be in class one times a
day and this remainder term is necessary
pple than any
so in class one which means
that the difference between
overall value of the series which is the
infinite some
into the sea
capital gains
partial sum is actually necessary both
us the south
capital in class one
how would that be proof well
i wish
summation here
in fascination the absolute value of the
summation is just a m plus one minus
eighty and close to class in the past
three months in trust for
insulin-dependent wherein starting
within even ornoff index
i've got an absolute value here so i
have to make sure that my result is
positive and i can't
force that by simply realizing that the
terms decrease or armani increasing
which means
this differences
non-negative this differences
non-negative
insulin
int so what
if i bracket that differently i get that
this is being plus one minus
in on the have number of mine is another
nine eight number mine is another monday
at number
so i realized
that this absolute value is necessary
prevent a_t_m_ plus one because all they
do as i keep subtracting
but i can't get videos zero because i
know that the result does not make it
so that means that the successor economy
and plus one
which is what we have claimed
that means if i
for example want to find the first for
decimal places off the value of the old
committee harmonic series
well that would mean
that would need to take my remainder
term
reach is there any harassment is
negatives want to be in plus one divided
by absolute value and make that
listener equal
then one over tend to the four th that
would give me a errors that
uh... most effects
but hostages in a
plus minus one
and that would mean that in his crate
articles and ten thousand
is about what they mean that i would
have to sit after at
the first two thousand terms office
series
well that gives me fast
uh... actual first
couple of decimal places are
approximately point six nine three zero
point six nine
three years
close to accurate right quick drop down
to
uh... two oh nine best
uh... basically then because the air
could affect institute in because there
was a shock those are just
recorded a couple more digits and that
would be
a pretty good estimate an estimate
wasn't tend to make it a four
the value
off the alternating
harmonic
cyrix
when in fact
we have some more
convergence tests for series as well as
some ways to actually estimate
values of series
should we choose to do so
you can already
anticipate what the homework is going to
be like you're going to have some
series for which you have to check
whether they're converted or diverted
inter-party current estimate some values
have fun with it
