Math 2300 Section 005 – Calculus II – Fall 2010
Notes on Taylor Series – October 18, 2010
Definition of a Taylor Series
The Taylor series for centered at is
Similarly, the Taylor series for centered at is
Examples
Since a Taylor series is the same as a Taylor polynomial, but is taken to infinite degree, we already know many Taylor series. From the last section, we know the following.
where
There is a bit of a technicality here. In section 10.1, we wrote between the functions and their Taylor polynomials. Now, we're writing instead. ( means “approximately equal to,” while means “equal to.”) The main point to understand here is detailed in the next portion of these notes.
Intervals of Convergence of Taylor Series
Here is the main point: Taylor series are power series. Remember that power series are given by a series in the form
where the are coefficients. Thus, for Taylor series, the derivative and factorial parts of each term account for the . But, more importantly, power series have a radius of convergence. This means that for some radius , the interval around where the series is centered will cause the series to converge. I.e., if is the center of the series then any in the interval plugged in to the series for will cause the series to converge. As to whether or not the specific numbers cause the series to converge, we have to explicitly test them and determine that for ourselves.
Like power series in general, Taylor series may have an infinite radius of convergence. This simply means that any real number value for will cause the series to converge.
Let's look at two examples, one with an infinite radius of convergence and one with a finite radius of convergence.
Examples of Radius/Interval of Convergence

Find the radius and interval of convergence of the Taylor series for centered at .
Solution: We know that the Taylor series for at is
Recall that finding the radius of convergence of a power series requires us to use the ratio test. Specifically, the ratio test says that, in order for the series to converge, we must have
How this relates to the radius of a power series isn't immediately obvious, but let's work through this calculation for the above Taylor series for and it will become more obvious along the way. Notice that in our situation we have . Thus, we need the following limit to be less than 1 in order for the series to converge.
The last step follows since and this has nothing to do with (it is essentially a constant as far as is concerned) and we may pull it to the front of the limit. Now, we're considering the limit
where we have replaced by since they are obviously equal. (Remember that is the distance between and , and so is a nice mathematical way to represent how far is from zero.) What we have shown is that for satisfying
the Taylor (power) series for evaluated at that will converge. Since the limit is zero, the inequality is true for all . This means that the Taylor series for converges for all real numbers . Thus, we have an infinite radius of convergence and the interval of convergence is .

Find the radius and interval of convergence of the Taylor series of centered at .
Solution: Since the Taylor series is
we have the following limit calculation given by the ratio test. Again, we just pick everything apart inside the limit until we can take the limit.
What the ratio test tells us is that the series will converge if the value of this limit calculation is less than 1. That is, the series converges when . Thus, the radius of convergence is 1. Therefore, we know that the series at least converges for within 1 of 0, i.e., on . What this doesn't tell us, unfortunately, is what happens at the endpoints of that interval. So, if we plug in or , we need to figure out whether the series will converge. This usually ends up being pretty easy. In this case, plugging in probably won't work, since the function isn't defined there. We'll plug it in to the series anyway and see what happens. We get
Since plugging in leads to a clearly nonconvergent series, we don't add to the interval of convergence and the interval therefore stays open at . However, plugging in we find that
and this converges by the alternating series test. Therefore, we add to the interval of convergence and our final interval of convergence is .
Binomial Series
Not all Taylor series are infinitely long. For instance, you know that the Taylor polynomials of any polynomial can only have up to the degree of the original polynomial. If you understand why Taylor polynomials of polynomials are limited by degree in this way, then you will understand that Taylor series of polynomials are the same.
Binomial series are Taylor series with a specific form. They are defined by
and when is some positive integer this just becomes
Examples of Binomial Series

Find the Taylor series of around .
Solution: This is really just asking us to expand the polynomial . We could actually expand it out, but that would take a really long time. Instead, using the Binomial series with we have

Find the Taylor series for about .
Solution: This is the Binomial series with . We find