Hi everyone.
Welcome back to integralcalc.com.
We’re going to be doing another infinite
series problem today.
This one involves perpetual income streams.
So it’s like an application of infinite
series which is really interesting.
I've written the formula for present value
here on the board and also the formula that
we use for infinite series.
So the question is: How much do we need right
now to establish a perpetual fund that pays
a thousand dollars at the end of every year
if the account pays five percent annually,
compounded monthly and we never add to it?
So perpetual fund means that forever into
eternity, this fund will pay us back a thousand
dollars at the end of every year.
It’s growing at a rate of five percent annual
interest and the interest is compounded monthly.
In other words, twelve times a year.
So let's go ahead first and plug that information
into our present value formula.
And the reason that we're using a present
value formula instead of a future value formula
is because it asks us how much we need to
invest right now.
So we're solving for the present value, what
we need to invest now.
We’re going to plug everything else into
the right side of the equation.
So let's say the present value equals, and
they want us to pay a stack of one thousand
dollars at the end of every year.
So one way that we can plug in this information,
we can say that we're going to get a thousand
dollars and that we'll just put in at the
end of one year.
So when we do that, leave the one.
They said that the rate which is r was five
percent a year so that's point zero five,
five percent.
n is the number of times the interest is compounded
each year and it said it was compounded monthly
so that's twelve times a year and so then
we have here, negative n times t, so negative
twelve again and then t, what we're gonna
do is go ahead and put in one because we're
looking for a thousand dollars to be distributed
back at the end of one year.
So we can go ahead and plug in a one there.
So that is the formula for the amount of money
that we need to establish in the fund right
now for it to pay us a thousand dollars back
at the end of the first year.
But we want this thousand dollars back at
the end of every year from now until forever,
so what that's going to look like is the same
thing we have here, a thousand times one plus
point zero five over twelve times negative
twelve, to the negative twelve times one.
But we also want them to pay us back at the
end of the second year and the third year
and the fourth year, so theoretically, we
could continue to write the same thing, to
the negative twelve times two, and this would
represent paying us back a thousand dollars
at the end of the second year as well and
we could continue and write what we're going
to get back at the end of the third year,
to the negative twelve times three (sorry
that's so crunched) but we could keep going
forever and that would be a representation
in infinite series form of the amount present
value that we need to establish now.
So you can see that this is turning into an
infinite series and now that we  recognize
that it's an infinite series, what we want
to do is use our infinite series formula to
go ahead and solve for pv, for present value.
So the infinite series formula that we always
use is first, if we have a series in the form
x to the k where the first value of k is zero
and the series is going to infinity and the
absolute value of x is less than one, then
the series converges to the value one over
one minus x, which right now sounds very confusing
but it's actually not.
We take this in three parts, let's just go
ahead and separate these in commas, we take
these series or this formula in three parts
and first we look at the first part.
We need our series to be in the form x to
the k.
So the way that we always try to do that is
factor our series.
So you can see that each term of the series
is the same except for this value of t here.
So we can factor our first term and in fact
you usually wanna factor out the first term,
that's the most common thing that you do,
so negative twelve times one.
So if we factor that out of the series, then
to get this first term here, we multiply by
one because it's the same thing.
And then what we multiply to get this second
term here?
Well, we don't need to multiply by a thousand
again because that was factored out here.
What we do need to multiply by is one plus
point zero five over twelve times negative
twelve only, because if we multiply this term
here by this whole thing out in front, we'll
have a thousand times one plus point zero
five to the twelve and then this is negative
twelve, and this is negative twelve, so it's
the same as negative twelve times two, which
is what we have up here.
So multiplying this by this gives us this
second term, the third term of course would
be one plus point zero five over twelve times
negative twelve times two, because this exponent
and this exponent multiplied together would
be negative twelve times three which is what
we want there.
And then of course this would continue forever.
So we factor the series with the goal of having
the series begin with one.
That's simplest way to go about that and that's
why we factored out exactly what this first
term was so that the first term in the series 
would be one, because one times this gives
you that first term.
So we  factor out the first term so that
the series begins with one because then it's
a dead giveaway that the second term right
here is our x and we want to find out what
x is so that we can then apply it to these
two parts of our formula.
So we're always looking at the second term
as x.
And it's, if you get the series to start with
one, by factoring something out, then it's
always the second term here immediately following
the one that should be your x here.
So if that's our x, which it is, then what
we need to do is look at the second part of
our formula and figure out if the absolute
value of x is less than one.
So if we actually do the math in our calculator,
one plus point zero five over twelve, that
whole thing raised to the negative twelve,
we actually get point nine five something
which is less than one, so we know that this
formula then applies.
The absolute value is less than one which
means that the series does converge.
All that means is that we can calculate the
value of the series which is good for us because
we need to find present value.
So since it does converge, we can use this
formula here, one over one minus x to determine
the present value of the series.
One over one minus x is the value of our series
and this is our x.
So present value is going to be one thousand
times one plus point zero five over twelve
to the negative twelve times one.
We have to leave this because this here only
represents this series, so only represents
these here in bracket so you got this multiplied
out in front, you have to leave it.
So we're going to do that but then we're gonna
multiply times one over one minus x, here
which represents the series so one plus point
zero five over twelve to the negative twelve,
right there.
So that should be our present value.
And when we do the math in our calculator
and that would be something that you need
to be very careful about.
I recommend starting from the inside here
and doing this in your calculator and then
working out instead of plugging the whole
thing in.
But when we plug all that in our calculator,
the value that we get for the series is nine
thousand five hundred and forty-five and eighty
cents which is the amount that we must invest
now for the fund to pay us a thousand dollars
at the end of every year forever.
