Hi everyone.
Welcome back to integralcalc.com.
We’re going to be doing another infinite
series problem today.
This one is an infinite decimal.
It’s zero point two one, repeating, that's
what this line mens.
So that's just the same as zero point two
one two one two one two one dot dot dot, forever.
That’s what this represents.
So, they've given us this infinite decimal
and they've asked us to write it as a simple
fraction, which means that we need to determine
what value it converts to or the sum of the
series.
And the formula that we used to determine
this, and the formula we used for almost all
the infinite series is the following.
It's a three part formula.
The first part is 
this right here.
All this means is the sum of the series x
to the k when the first value of k is zero
and the series goes on to infinity.
The second part of the formula is the absolute
value of x less than one.
If the absolute value of x is less than one,
this x here is less than one, the series converges.
If it's greater than or equal to one, it diverges.
But that's where we stop.
If it diverges, we're done.
So it diverges and we're done with it.
If it converges, then we can find its sum
which we use one over one minus x to find.
And again these three x's here are the same
in each case.
So once we find this x, then we can apply
it to these two parts of the formula so the
first thing that we have to do with this series
here is figure out whether or not it takes
this form if it can be represented by this
form here, meaning x to the k.
So the first thing we're going to do is turn
this decimal into a fraction.
And the way that we would do that, we’re
going to take each part separately, like this.
The first point two one, (right?) since this
one is in the hundredths place, that’s,
this first term here is the same thing as
twenty one over one hundred, so we'll write
twenty one over one hundred.
And then this second one, this one is in the
ten thousandths place so it would be twenty
one over ten thousand, right?
We just added two zeros here to this.
Then the third twenty over, we've got one
and then one, two, three, four and then add
two zeros.
So that would be the one millionth place and
so we could go on forever.
So that's the first step.
We write this as a fraction.
You always do this based on where the first
section, whatever decimal place it ends.
This being the hundredths place, ten thousandths
place, one millionths place, that's how you
know what to put in the bottom here.
So once you've done that, then you can factor
the series.
And the way that you want to do that is by
taking out the first term here.
You always want to just factor out the first
term.
So in this case our first term is twenty one
over one hundred, so we want to say twenty
one over one hundred times one for this term
plus one over, and that would be one hundred,
because four zeros.
So one over one hundred times twenty one over
one hundred is the same thing as twenty one
over ten thousand here.
So I'm just factoring the series and then
we have one over we'll have four zeros, so
that's one over ten thousand plus dot dot
dot.
Okay, so we’ve factored the series.
That’s the first important thing that we
need to do.
Second thing that we need to do now that the
series is factored, is determine, once we
have the series like this where you want to
get to the point you want to get to, is where
you have the first term in your series equal
to one.
You want the series to start with one, which
is why we've factored out the twenty one over
one hundred.
The sole purpose of taking this twenty one
over one hundred out is that so we could get
this series to start with one because once
the series starts with one, we then know almost
certainly that this second term right here,
the term immediately following the one is
our x term here, is equal to x.
The way that we know that, the  way that
we can double check it and that we can prove
that we're right, if we expand this series
here, the way this series  is actually written
out, if we expand it is the following, x to
the zero, right?
Because the first value of k is zero so the
first exponent on the x is zero and since
it goes on into infinity, it goes plus x to
the one plus x to the two plus x to the three
plus dot dot dot.
That’s what this series here looks like
when we expand it.
So you can see if we simplify this, anythng
raised to the zero power, no matter what it
is is equal to one so x to the zero is the
same thing as one, is equal to one.
x to the one, that's for, that we never write
anything to the one power, to the power of
one, that's x so we have one plus x plus x
squared plus  x cubed plus dot dot dot.
So we've simplified this series.
So you can see here, right?
When we get this series to start with one,
it most closely approximates this series because
we have the one here in the first term and
then the reason I'm saying this is x is because
the second term here is x so we're hoping
that this is x.
This of course then would be x squared, the
next term would be x cubed.
So that's why we factored this so that we
could get the series to start with one because
then whatever this term is, is most likely
x.
So let's go ahead and just double check.
If x is one over one hundred then our series,
we're just going to go ahead and apply it
here.
Our series is one plus one over one hundred
plus one over one hundred squared, and this
is actually one over one hundred and this
whole thing would be squared, right?
Because the whole x is squared plus one over
one hundred cubed plus dot dot dot.
So that would be, if one over hundred is our
x, then that's what the series would look
like.
So what we need to do is make sure that this
series is the same thing as this series here.
And we have the one, we have the one over
one hundred, so the question is is one over
one hundred squared the same as one over ten
thousand and in fact it is, right?
One over one hundred squared would be one
over one with four zeros, which is ten thousand,
one over ten thousand.
So the series is matching up.
We can see that they're going to end up being
the same series so we've proven, we went on
with one over one hundred being our x term
as a hunch because when you have the series
staring with one, the second term is almost
always your x, but we double checked by plugging
one over one hundred back into this series
right here to make sure that this was matching
this.
So one over one hundred is our x term.
Make sure when you have a series like this
and you have to factor out the first term
to get this series, you don't forget about
this term because you're answer will be wrong
if you forget about it.
So you have the twenty one over one hundred
and then one over one hundred is your x.
So the way that we write that, our series,
this infinite decimal is represented like
the following.
Since we have twenty one over one hundred,
multiplied by the whole series, we have twenty
one over one hundred multiplied by the whole
series, the series represented of course,
remember we said our x was one over one hundred?
So we plug one over one hundred here for x
and keep everything else.
And we have now re-written this infinite decimal
in this form so that this infinite decimal
is written in the form of this formula here.
This is the formula and we know what our x
is so that we can determine this sum and find
out what I've not converted.
So if our x is one over one hundred, now we
need to plug that in here.
The absolute value of one over one hundred
is less than one.
We actually need to find out if it is or not.
Absolute value just means take the positive
value of whatever is the positive value of
the inside here.
So if this were negative one over one hundred,
you would take the positive values so you
would just take away that negative sign and
it would be one over one hundred.
Since our value is already positive, we don't
really have to worry about it and the absolute
value brackets really have no effect.
So our question is one over one hundred less
than one?
And of course we know that it is.
So since this is less than one, our series
does converge, converges and then since it
does converge, we need to find the sum using
the formula right here, one over one minus
x.
Since our x is one over one hundred, we go
ahead and say one over one minus one over
one hundred but we can't forget to not forget
about our twenty one over one hundred.
Since that is multiplied by the whole series,
we have to multiply it out in front here.
So now let's go ahead and simplify this.
The first thing I'm going to do is make this
one, one hundred over one hundred so that
I can, right, one over one hundred is the
same thing as one, but I want to simplify
this denominator here and I need common denominators,
this one over one hundred to match so that
I can simplify.
So I'll have twenty one over one hundred times
one over one hundred minus one is ninety nine
in the numerator and then of course, since
I have one hundred in the bottom of both fractions,
I can combine those two and then I have one
divided by a fraction.
Whenever I have something divided by a fraction,
instead of dividing here, I can multiply by
the inverse.
So this one over ninety nine one hundredths
is the same thing as (we're going to write
twenty one over one hundred), is the same
thing as one times one hundred over ninety
nine.
I flipped this fraction upside down and multiplied
these two together instead of dividing which
is a common algebraic operation that you should
definitely get comfortable with.
Now we've got one times one hundred over ninety
nine which of course is just the same thing
as one hundred over ninety nine.
So now we have these two fractions multiplied
together, you can see we have one hundred
on the bottom and a one hundred on the top
which means those two cancels and we're left
with twenty one over ninety nine.
So that is our final answer.
The series converges to twenty one over ninety
nine.
Twenty one over ninety nine is the sum of
the series and the answer that we would write
down is converges to twenty one over ninety
nine.
Thanks, guys.
See you next time!
