Welcome to Calculus,
I'm Professor Ghreist, we're about to begin
Lecture 54 on power series. >> At last, we've assembles all
the tools we need to adequately and accurately describe how, when, and
in what manner infinite series converge. In this lesson, we're going to
apply this tools to power series. >> In this chapter, and indeed in this
course, we've seen several instances of turning functions into sequences
through some sampling process. We now turn to the problem
of reversing that machine, of converting sequences into functions. One mechanism for doing so
is the power series. A power series in x is
simply a series that has the variable x inside of
it as a monomial term. Something of the from sum and
goes from 0 to infinity. Of a sub n, x to the n, where now
a sub n is considered a coefficient. We're going to think of the notion
of a power series as a machine or an operator that converts a sequence, a sub n, into a function, f of x. Let's begin with some simple examples. If we have a sequence,
all of whose terms are 0, except for one in the nth slot, then the power series is simply
the monomial x to the n. Now if we have a finite sequence, in the sense that only finitely
many of the terms are non 0, then this will have a power series
that is a sum of monomials with coefficients in front of them
that eventually terminates. We call that a polynomial. Now, if we have something that
does not eventually end in 0s, we might have to be a bit more careful,
but this isn't so mysterious. If we have the sequence of all 1s,
we recognize that the power series for that sequence is simply
the geometric series. And this gives us the function 1 over 1-x, at least when x is less than 1,
an absolute value. Other sequences can be easily seen to
correspond to interesting functions, though one may have to do
a bit of algebra to see it. If we look at certain sequences,
we'll recognize some old friends. Log of one+x, or e to the x. These correspond to certain power series. Now, what if we have a sequence
that does not go to 0. Say, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc. Will that converge to anything? Well, of course, this isn't so bad. This one corresponds to the function
1 over quantity 1+x squared, as you can see from squaring
the geometric series. In fact, we can have sequences
that grow rather large but which have power series that
converge to something very simple. Now, what if we just take any old
interesting sequence of coefficients. What will that converge to? Sometimes it's a bit of a mystery. Let's look at the Fibonacci sequence, and let us denote by script F the power
series whose coefficients are F sub n, the terms in the Fibonacci sequence. Now, if we were to multiply this
power series by x what would happen? It would give us a new power
series associated to a shift, a right shift, of the Fibonacci Sequence. If we were to do it again, then we would
shift over to the right yet once more. And the reason for doing this
is to line up the coefficient to match the recursion relation that
the Fibonacci Sequence satisfies. Fn+2 = Fn + 1 + Fn. Now if we take a look at the coefficients
in front of the monomial term, x to the n, for these three power
series that we have written down. Then we see from the recursion relation
the relationship between them. The first is the sum of the latter two. And so if we were to subtract
x times script F and x squared times script F from script F,
what would we obtain? Most of the coefficients of
the monomial terms are zero, because of the recursion relation. When we add up these power series,
the only thing that is non-zero is in the constant term or
we get F naught and the first order term where
we get F1x- F naught x. Now because we know the exact
values of F naught and F1, we see that, on the left,
we have F- xF- x squared F. On the right, we have simply, x. Solving for
script F gives us the power series formula that script F, the power series, the Fibonacci sequence is equal
to x over 1- x- x squared. Now, that's what it is,
but what does it mean? What do we do with this? These functions that generate
the power series have within them, all sorts of interesting questions. They tie together recursive properties,
enumerative properties, approximation properties and convergence. We're going to focus on
issues of convergence. Consider a power series F of x,
sum of a sub N, x to the n. The following theorem holds. For some capital R between 0 and infinity, perhaps achieving either
of those values f of x has the following convergence behavior. It converges absolutely,
if x is less than R in absolute value, it diverges if x is bigger
than R in absolute value. Now this R is a very special number
associated to the power series. It's called the radius of convergence. Let's see,
how would we prove this theorem? What tool would we turn to,
to determine convergence? Well, let's start with our friend,
the ratio test. Here, what we need to do is compute rho. The limit is n goes to
infinity of the ratio, the n plus first term to the nth term. Now, what are the terms? Well, here they are a sub
n times x to the n. So taking the ratio of incident
terms will tell us what we need. Now, we have to be careful
to take the absolute values since the ratio test requires
a positive sequence. Okay, so doing that what do we see? We get some cancellation with the xs and
we can rewrite this as the limit as n goes to infinity of
the absolute value of a sub n + 1 over the absolute value of a sub n
times the absolute value of x. Now, if that is less than 1,
we have absolute convergence. If that is bigger than 1,
we have divergence. So where is this capital R coming from? Well, it is precisely the reciprocal
of this limiting coefficient. Since if we divide by this on both sides, we get the appropriate answer. When rho is less than 1,
that's the same thing as saying that x is strictly less
than R in absolute value. That means we have absolute convergence. When rho is bigger than 1 that
is the same thing as saying x is greater than R in absolute value,
and that means divergence. So what matters is this radius
of convergence capital R. The limit as n goes to infinity of
absolute a sub n over a sub n+1. Be sure to remember that, and
be sure to remember that it's the nth term over the n plus first term,
not the other way. Now, this Radius of convergence is
called a radius of convergence because it's telling you the distance
away from 0 past which your power series
is going to diverge. But within that, you're fine. Now, you do have to be careful,
we've said nothing about the end points of the domain, because of course, the ratio
test tells us nothing in that case. So you're going to have to be careful and
check explicitly at those end points. Let's see a simple example
if we consider f of x. The sum over n -1 to the n,
square root of n, quantity 2x to the n. Then our coefficient
a sub n is -1 to the n, square root of n, 2 to the n. To compute the radius of convergence,
we take the limit of the a sub n divided by a sub
n + 1 all in absolute value. That gives us the limit as n goes
to infinity of square root of n, 2 to the n, over square root of n + 1,
2 to the n + 1. That is, as you can easily see, one-half. Therefore, within one-half of 0,
we have absolute convergence. Outside of that, we have divergence. What happens at the end points? Well, we need to check what happens
at x = -one-half if we plug in a value of -one-half in 4x,
what do we get? We get the sum over n of square root of n. Since the -1 to the ns cancel. That is a divergent series. So at the left hand end
point we have divergence. At the right hand end point,
at x= one-half, you can easily see that we get
the same sort of series that, too, is going to diverge
by the nth term test. Now other series that we've seen
throughout this course have similar behavior. If we look at the geometric series, this is the power series where all
the coefficients are equal to one. The radius of convergence
in this case equals 1, and this diverges at both
endpoints at plus and minus 1. If on the other hand we consider the power
series with coefficient 1 over n, then goes from 1 to infinity. Then what do we see? This, too, has a radius of
convergence equal to 1, and at the right hand end point we obtain the
harmonic series, which certainly diverges. But at the left-hand end point,
that x = -1, we get the alternating harmonic series. That converges, but
it does so conditionally, not absolutely. If we consider the power series
x to the n over n squared, where the coefficient is 1 over n squared, this also has radius of
convergence equal to 1. But it converges at both end points and does so, therefore, absolutely. Some power series come to us
in a slightly different form, that of a shifted power series,
something of the form sum over n of a sub n times quantity x- c to the nth. This of course is just a power
series shifted over by c units. In this case, if we compute the exact
same radius of convergence, then what we will find is that this
shifted power series converges when x-c is less than R in absolute value. That is when x is within R of c. R is thus a distance to the center,
c, of the shifter power series. For an example of a shifted power series, consider the sum of 3x-2 to
the n over n times 4 to the n. Now this is not in the form
of x-c to the n, so we're going to have to do a little bit
of algebra to get it into that form. Claim that by factoring out
a three-quarters to the n, and absorbing that 3 to the n term from
the denominator into the numerator. We get the equivalent
sum of 1 over n times three-quarters to the n times
x -two-thirds to the n. That means that this shifted power
series is centered, and x = two-thirds. Now to determine
the radius of convergence, we isolate the coefficient a sub n as 1
over n times three-quarters to the n. We compute the radius R is
the limit then goes to infinity of a sub n over a sub n + 1. That yields, after some algebra,
the limit of n + 1 over n times four-thirds, which is
precisely four-thirds. Therefore, the domain of
absolute convergence is all x within four-thirds of
two-thirds the center. Now, what happens at the boundary points? Well, on the left,
we have x = -two-thirds. That is, two-thirds minus four-thirds. Substituting in x = -two-thirds,
we obtain, after a little bit of cancellation,
the alternating harmonic series. That, of course, converges. When we substitute in the right
hand end point, that is, x = two-thirds + four-thirds or 2. Then we obtain, substituting in
a series which is the harmonic series. This, of course, diverges, so that we have conditional convergence
on the left, divergence on the right. >> In our next lesson we're
going to focus attention on a particular type of power series
obtained from a smooth function f of x. Perhaps you've heard of these,
they're called Taylor series.