Hi everyone.
Welcome back to integralcalc.com.
We’re doing another infinite series problem
today.
This one is the following.
We have three halves minus three fourths plus
three eighths minus three sixteenths plus
dot dot dot.
So this is the series they gave us and they've
asked us to determine whether or not the series
converges and if it does converge, to find
the sum of the series.
So given those instructions, for almost all
infinite series problems, the formula that
we use is the following three-part formula.
I’m going to write it down e to the three
parts.
The first part is x to the k, infinity, absolute
value x less than one and one over one minus
x.
So I hope you guys can see that.
So this formula, this is the first part, the
second part, the third part.
The first part means the sum of the infinite
series, we know its infinite because we have
infinity up here, of the series that takes
the form x to the k where the first value
of k in the series is zero and the series
goes on into infinity.
If the series takes that form then we can
apply this two, this being if the absolute
value of x is less than one, then the series
converges.
If the absolute value of x is greater or equal
to one, then the series diverges and we can
finish the problem, and if the series diverges,
we can't find the sum of the series.
It’s indeterminate so we're done.
We just write diverges.
But if it does converge, meaning if the absolute
value of x is less than one, then the series
converges then we set the formula one over
one mins x to find the sum of the series,
or the value that the series converges to.
And of course these three x's are the same
x, the same value.
So the first thing we need to do is figure
out whether or not our series can be written
in this form and if it can, we need to find
out what x is because we need x to plug it
in to these two parts of the formula to finish
the prblem.
So the first thing that we do to figure our
whether or not our series takes this form
here, is when we're given a series like this
and it's written out, we've got fractions,
the first thing we wanna do is to factor out
the first term in the series.
It's the simplest way of going about.
So we're going to factor out three halves,
so if we factor out three halves, of course
the first term there is one, three halves
times three halves is one.
So then we would have minus one over two,
right, cause three times one is three and
two times two is four, then we have plus one
over four, and then we have minus one over
eight plus dot dot dot.
Okay, so we factor out the first term in this
series.
The reason we want to do that is because above
anything else, what we want to do is to make
sure that our series begins with one, that
the first term in our series is one.
So we took out the three halves so we could
get it to start with one.
So we're already in good shape.
What we need to do though now is we need to
determine if this series here starting with
one, going on into infinity is in the same
form as this series.
So what I like to do is write this series
out so that we can see it.
So this series would start with x to the zero
because the first value of k in this series
is zero so we plug in zero for k first.
Then since it goes on into infinity, we say
plus x to the one, plus x to the two plus
x to the three plus dot dot dot.
This is this series written out, expanded.
This is the abbreviation, the condensed version,
the representation of this series.
So this series, if we now simplify it, right?
Anything raised to the zero power is one so
we have one here and then x to the one, we
can just write as x plus x squared plus x
cubed pus dot dot dot.
So you can see, when we simplified this series
right, the reason that I wanted to factor
out the three halves, so that the series would
begin with one is because this series begins
with one when we expand this series, we write
it out, we simplify it, this series begins
with one, so both of our series begin with
one then we're closer to having a math job
and figuring out what x is.
So now you can see, I'm going to go ahead
and erase this because we don't need this
line here.
Now you can see that we've expanded this series,
right?
Both of these series begin with one.
Remember the whole goal of going through this
exercise is to figure out what x is.
So that we can then apply it to these two
parts of the formula right?
If your series both begin with one like this,
here's our x here, it's the second term, I'm
guessing that our x is gonna be this negative
one half here.
You always have to include the sign here otherwise
it won't work.
So if it's positive sign, that's not a big
deal, it's just one half but if that's a negative
sign, then x is negative one-half because
if you plugged in negative one-half here right,
you've got a negative.
So and this is pretty much a dead giveaway.
Once you get your series into this form, it
starts with one, almost all of the time, you
can guess that the second term, the one immediately
following the one here is your x, because
this is the expanded series and chances are
they're going to match.
x is the second term and your second term
is negative one-half.
So we can always bet that our x is negative
one-half.
But let's just plug negative one-half into
this series to make sure that it's going to
match up with this series here.
So if we plug negative one-half and you know
what, we actually need to, well we need to
do it this way.
If we plug in negative one-half into our series,
we get one plus negative one-half plus negative
one half squared plus negative one-half cubed
plus dot dot dot.
Okay, so that's our series.
We can just go ahead and erase this stuff.
So let's go ahead and now simplify this, right?
We'll get one and then we have plus a negative
so we get minus one-half and then we have
a negative one-half squared which is a positive,
a negative times a negative which is positive,
positive one fourth so plus one-fourth and
then negative one-half cubed.
A negative sign cubed is still a negative
sign and we have, two times two is four times
two is eight on the bottom so we have minus
one eigth and then plus dot dot dot.
So you can see, look our series, this one
and this one, are identical.
They do match, which means that we just confirmed
that negative one-half is our value for x.
We guessed that it was because it was the
first term after the one, which is almost
always a dead giveaway.
You should guess that and make sure to include
the punctuation or the sign here and we confirmed
it because we plugged that negative one-half
back into our expanded series here.
And we see that they match so we know that
we're right and negative one-half is our x
value.
So that, we went through all of that just
to find out what x is.
We know that our series takes this form because
we got the two to match up and we know that
our x is negative one-half.
So now, remember, we had that three-halves
out in front where our series start with one
and then went on, that was the representation
of this series here but we have that three-halves
out in front and we can't just forget about
that. 
We have to keep it otherwise our answer won't
come out right.
So the way that we write our series now is
negative one-half to the k.
So we said that our x was negative one-half
so we go ahead and plug it in here.
But we have to have that three-halves out
in front multiplied by this whole series.
So this would be the abbreviated version or
the condensed version of our whole series
here.
This is how you would write it in its summation
notation with the three-halves out in front
and then the negative one-half plugged in
for x.
So now that we have our series condensed into
this form, we can go ahead and take our x
which is negative one-half and plug it in
these two components to see whether or not
the series converges and if it does, what
the sum of the series is.
So let's go ahead and plug negative one half
into or absolute value brackets over there
for our convergence test, right?
Absolute value turns this into one-half.
Absolute value just basically means, whatever
sign's inside of this, makes it a positive
so since this is a negative one-half, the
absolute value is one-half.
The absolute value of negative eleven would
be positive eleven. 
The absolute value of negative one point three
seven five would be positive one point three
seven five, so you just take away the negative
sign.
So we've got positive one-half, is that less
than one?
Of course it is so since it's less than one,
the series does converge.
If this had been greater than or equal to
one, the series would diverge and we would
just write diverges and be done.
But since it's less than one, that means it
does converge, and since it converges, that
means we have to take the problem one step
further and use this part here to find the
sum of the series.
So again, we plugged in negative one-half
and we get one minus negative one-half there
for x and we -an go and simplify this.
We get one over one plus one-half because
we have minus a negative, so we make that
a positive.
Let’s go ahead and make this one here, we
can write that as two over two, right?
Two over two is the same thing as one.
I'm making it two over two so that I could
combine these two fractions here that now
have a common denominator so I have one over
two plus one is three and then common denominator
of two and now I have one over three-halves
which is the same thing as one times the inverse
of this fraction, which is two thirds.
And so we've got one times two-thirds which
is just two-thirds.
But remember we had this three-halves out
in-front here, so when we plug in the negative
one-half to this formula here, that's not
our final answer.
We still have to multiply by this three halves
here and what we should actually do is put
that out in front at the beginning, put that
three halves out in front there at the beginning
and go through, bring it through all of the
steps instead of just adding it to the end
so just you don't forget.
But we have to multiply this by three-halves.
So you can see we have three over three and
two over two which is just gonna, we're just
going to write it out.
It’s just six over six, so the threes cancel,
the twos cancel.
And this just simplifies to one.
So the series converges to one or the sum
of the series is equal to one, same thing.
But that is how we determine the sum of the
series so you can see our series which can
also be written in this form here in summation
notation like this.
This series converges based on the convergence
test to the value of one which we got from
this part of the formula here.
So that's our final answer, guys.
See you next time.
