Power series manipulation examples

[Sean Fitzpatrick]

One thing that I've noticed we don't have a lot of examples for is manipulation of power series.

This is a skill that comes up a lot in physics as well as later math courses like complex analysis, so I think a bit more practice would be good.

The types of problem I'm thinking of here are ones like finding a power series for the function

f(x) = x^3/(4+x^2)

where we manipulate to get 

f(x) = (x^3/4)*(1/(1-(-x^2/4)))

and then geometric series formula gives the sum of (-1)^nx^(2n)/4^n for the factor on the right, and multiplying in the x^3/4 gives the power series representation

f(x) = sum_(n=0)^infinity (-1)^n x^(2n+3)/4^(n+1)

What do you think about adding some examples/exercises like this?

[Rob Beezer]

On 3/25/21 12:12 PM, Sean Fitzpatrick wrote:
> This is a skill that comes up a lot in physics as well as later math courses
> like complex analysis, so I think a bit more practice would be good.

Also super useful for generating functions in combinatorics.

[Sean Fitzpatrick]

Oh yeah! It's been about 2 decades since I took combinatorics. From a prof who liked to do things by translating each problem into a regular expression and then applying a "functor" to produce the generating function.

[gregory...@gmail.com]

Not opposed to it. May be fun to include an example/exercise that relates to generating functions. (I don't want to get into the theory of generating functions; rather, just make reference to "these coefficients are useful when solving counting/combinatorics problems" or something like that.

[Sean Fitzpatrick]

I'm going to start working on this shortly. A couple of questions, mostly for Greg:

1. Does this belong in the section on power series, or Taylor series?

2. In the Taylor series section, there are a couple of items I'd like to update.

First, Theorem 10.7.12: Algebra of Power Series

The composition result is problematic: if h(x) is not polynomial, then plugging h(x) into the power series for f(x) does not produce a power series.

I would suggest restricting this to only the case that h(x) is a monomial (and perhaps mention in an aside that polynomials can work, but it's messy).

Actually, I believe it's true that we can plug the power series for h(x) into the power series for f(x), and manipulate to get a power series equal to the Taylor series for f(h(x)).

(One should perhaps note that we have to be mindful of the centre: to use the power series of h(x), centred at x=c, we need to use the power series of f(x), centred at x=h(c).

But this is maybe more technical than we'd like to get.)

Second, the example with ln(sqrt(x)): I don't like that we plug sqrt(x) into the power series for ln(x), and then say "well, this is not strictly a power series..."

I would rather:

a) note that ln(sqrt(x)) = 0.5 ln(x), and then just use the power series for ln(x)

or

b) plug the power series for sqrt(x), also centred at x=1 (conveniently, sqrt(1)=1, so we can do this), into the power series for ln(x)

or even:

c) see if we can compare both of these approaches.

[David Farmer]

Plugging a power series into another power series is just
fine, provided both have a positive radius of convergence
(which is the case with all examples from intro calculus).

I prefer to say "power series" when the center is at 0,
which is the sense I meant above. When the center is
different, I say "Taylor series centered at...". When you
compose Taylor series with different centers, you need the
inner center to be within the radius of convergence of
the outer series.

Maybe the phrase "is not strictly a power series," could be
rewritten "is not written in the form of a power series"?

That is a great example of why the technical condition in the
2nd paragraph above is needed: You cannot expand and collect
terms in that series for log(sqrt(x)) to get a power series.

If you try to do that, the constant term in the series is

-1 - 1/2 - 1/3 - 1/4 - ...

which diverges. Could be worth making into an exploration
exercise.

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[Sean Fitzpatrick]

That's a good idea. I was just playing around with this example.

Neither ln(x) and sqrt(x) have power series at 0, for obvious reasons. Usually we're given ln(1+x) and sqrt(1+x) instead.

Or you can do Taylor series (for both) centred at x=1. These you can compose:

ln(x) = (x-1)-1/2(x-2)^2+1/3(x-1)^3-...

sqrt(x) = 1+1/2(x-1)-1/4(x-1)^2+1/16(x-1)^3-...

To do ln(sqrt(x)), you have powers of (sqrt(x)-1), which is not a power series. But by the above,

sqrt(x)-1 = 1/2(x-1)-1/4(x-1)^2+1/16(x--1)^3-...

If you plug this in for (x-1) in the power series for ln(x), truncate everything higher than (x-1)^3, and collect terms, you get

ln(sqrt(x)) = 1/2(x-1)-1/4(x-2)^2+1/6(x-1)^3-... = 1/2[(x-1)-1/2(x-1)^2+1/3(x-1)^3-...] = 1/2ln(x).

[Alex Jordan]

I use "power series" and "Taylor series" based on whether the
coefficients came from derivatives of some specific f being evaluated
at a. In other words, you can have some power series where there are
coefficients. Or you can have "the Taylor series for f centered at a"
in the special case that the coefficients are defined as f^{(n)}(a)/n!
for some f that has been specified.

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[gregory...@gmail.com]

1. I think this theorem belongs with Taylor Series. (To perhaps state the obvious) Alex's distinction is good; we introduce Power Series as a new mathematical object and the coefficients are just provided without context. In the Taylor Series section, the coefficients come from a source that's important to us; we are producing a different representation of a known/important function. In this context, we'd be interested in combining known functions together to get another known function. In the Power Series section, we *can* combine series, but we don't have a good motivation as to why we care.

2. I intentionally ignored the possibility of the series being centered anywhere but 0. Things got too messy. And to avoid different radii of convergence, I just stated that both had to converge for |x| < R (though perhaps they'd converge for larger R).

Two options for the composition of functions & h(x):

   a) Restrict h(x) to polynomials? What I was really after was "If I know the series for e^x, I should be able to figure out the series for e^{2x^2+1}."

   b) Perhaps the thm should be rewritten to have the composition be f(g(x)), where it's a composition of power series. That looks messy and we'd need an example.

I'd prefer a), and we could make a comment about being able to extend the truth of polynomials to series, but not actually doing it.

The ln(sqrt(x)) is problematic. One issue is that ln's series isn't centered at x=0, as the thm requires. I am violating a principle of giving simple, straightforward examples of the application of thms. I think I'd prefer to just replace that example with something that doesn't cause trouble. Then this example can be moved to a guided Exercise.

[Sean Fitzpatrick]

One way to keep the example would be to replace ln(x) with ln(1+x), and sqrt(x) with sqrt(1+x),
but I'm not sure that's any cleaner.

By the way, if you are free tomorrow afternoon, there's the weekly PreTeXt drop-in workshop.
I was planning on attending at least part of it. It runs 12-3 Pacific (3-6 Eastern) and I can be there for any part of it. I can forward you the notice from the PreTeXt development mailing list.

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[gregory...@gmail.com]

Sorry, Sean, I just now read this (Saturday morning), obviously missing your invitation.

Here's a concrete suggestion for how to edit the content in question.

1. Rewrite the theorem so that h(x) is a polynomial. Then f(h(x)) is clearly a power series. This is an overly restrictive condition on h(x), but it suffices for our needs.

2. Change Example 9.8.15. It currently has two parts, computing series for sin(x^2) and ln(sqrt(x)). Drop the second part; the first part is fine as x^2, the ''inside function'' of the composition, is a polynomial.

3. Create a new example that uses composition but has a different purpose. Approximate e^x with 1 + x + x^2/2 + x^3/6, and sin(x) with x - x^3/6 + x^5/120 (i.e., reference the first few terms already given for these series in a Key Idea), then use these to approximate e^sin(x) with a polynomial. With words, we explain that we are using the theorem that allows composition of polynomials into a series, along with the notion that these are approximations, and hence our result will be an approximation of the true series. 

We can gloss over the algebra, giving the final poly (with degree 15). We can also provide a sketch on [-1,1], where the approximation is quite good. 

And then address the "why do we care?" question at the end of the example. After all, we have computers that can calculate e^sin(x) for us; why all this algebra for an approximation? And the answer: this illustrates that approximations can work well, and we can use series approximations when *we don't know the actual function.* Computers can do the algebra quickly; we can likely find a useful approximation at the end. We use this principle in the 2 following examples, 9.8.16 & .17.

4. Change Exercise 32 of that section, which seeks an approximation to an integral of cos(sqrt(x)). Change it to cos(x^3).

[Sean Fitzpatrick]

This is now done at https://github.com/APEXCalculus/APEXCalculusPTX/pull/136/commits/acaedda052a97586be9cb34c4d98d04fb3e6bb8c

One part you may wish to remove: just for fun, I left in an example that discusses what would happen if you did (foolishly) go ahead and try to use Taylor series composition on ln(sqrt(x)), with each series centred at x=1.