Professor Dave here, let’s check out power series.
We’ve seen a few different types of series up until this point, so let’s check out another.
A power series is one that takes the form C sub N times X to the N power, from zero
to infinity.
So we get C zero first, with no X, because X to the zero is one.
Then the next term is C one times X.
Then C two times X squared, and so forth.
The C’s are constants that we call coefficients, and the sum of this series can be represented
by a function, F of X, where we just add up all the terms, resulting in what looks like
a polynomial with infinitely many terms.
The domain of this function is the set of all X values for which this series converges.
Now let’s look at some specific examples, and see what we can determine regarding convergence
and divergence.
Let’s say we take this general form and we make all the coefficients equal to one.
That would give us this geometric series here, which will be convergent when X is in between
negative one and one.
We may also see power series in this form, where the binomial X minus A is being raised
to the N power.
How can we assess whether this kind of power series is convergent or divergent?
Sometimes we will use the ratio test.
Take something like the quantity X minus three raised to the N power, over N. Remembering
the ratio test from the previous tutorial, let’s change N into N plus one, and then
take the regular version, and flip it to bring it up here.
To simplify, X minus three raised to the N plus one can become X minus three raised to
the N times X minus three raised to the one.
This more complicated term then cancels, leaving us with X minus three times N over N plus one.
Now let’s divide this fraction by N, to get one over the quantity one plus (one over N).
As these are all positive, we can pull them out of the absolute value brackets, leaving
just X minus three in there.
Now, bringing N up to infinity, this whole part just becomes one, leaving us with the
absolute value of X minus three.
In order to be convergent, this has to be less than one, so given the absolute value,
we know that X minus three must be less than one and greater than negative one.
Adding three throughout, we get X as being between two and four, so those are the values
for which the series converges.
In general, there is a theorem that summarizes the three possibilities that can describe
a power series in this form, depending on the precise series.
One possibility is that the series only converges when X equals A. A second possibility is that
the series converges for any value of X.
And lastly, there could be some positive number R such that if the absolute value of X minus
A is less than R, the series will converge, and if greater than R, the series will diverge,
just like the example we just completed.
In this third case, that number R is called the radius of convergence for that power series.
The first two cases also involve a radius of convergence, it is simply that for the
first case, the radius of convergence is zero, as there is only one value that will allow
for convergence, and for the second case, the radius of convergence is infinite, as
any value will allow for convergence.
We can also describe the interval of convergence as the interval describing all the values
of X for which the series converges.
Let’s try an example and see how this applies.
Take X to the N, over N factorial.
What can we do with this?
Let’s try the ratio test.
We get X to the N plus one over N plus one factorial, and then given the original series
in the denominator, we just multiply by the reciprocal, or N factorial over X to the N.
We’ve simplified something like this a couple times by now, so let’s change N plus one
factorial into N plus one times N factorial, and let’s change X to the N plus one into
X to the N times X to the one.
Most of this cancels out, and we are left with X over N plus one.
Finding the limit as N approaches infinity, we see that we will get zero no matter what
the value for X.
This means that the radius of convergence is infinity, and the interval of convergence
is negative infinity to positive infinity.
This fits the second possibility from that theorem, which tells us that this series is
always convergent.
That covers the basics regarding power series, so let’s check comprehension.