Error estimate for integral test

[Sean Fitzpatrick]

One of our instructors noted that we are missing the remainder estimate for series using the integral test. (This could be a case of, "it's in Stewart, and I'm using my lecture notes from 1993".)

It's a pretty straightforward consequence of the argument used to establish the integral test in the first place. Looking at upper and lower Riemann sums, if f(x) is a function for which the integral test applies, then

$$\int_{n+1}^\infty f(x) dx \leq \sum_{k=n+1}^\infty f(k) \leq \int_n^\infty f(x) dx$$

Is this something worth including in APEX?

[gregory...@gmail.com]

I'm a bit conflicted on this one. On one hand, sure - the result follows from statements already in the text. We could put it at the end of the Integral Test portion of that section, right before the Direct Comparison Test. It is a kind of nice result; an ambiguity that plagues all series is "how many terms is enough to give me a good approximation?" (especially knowing the harmonic series diverges, but oh-so-slowly). 

On the other hand: how applicable is this knowledge? The series must be decreasing, positive, and something we can actually integrate. (So forget using factorials.) And with technology, it isn't hard to get a good approximation very quickly along with a good sense of the error. Or even the exact answer.

What do you think?

[Alex Jordan]

Let me first say that I have no opinion on adding this to the book.

In a course where I've gone all out with estimation techniques, I'd
counter one of Greg's points. If those integrals in the bounds are
like
\int_{n}^{∞} 1/[x^5+1] dx
(too hard to antidifferentiate)

I would show students that this improper integral can become proper
with a substitution like:
x = 1/u
which gives
\int_{0}^{1/n} u^3/[u^5+1] du

which is still too hard to antidifferentiate, but now that the
integral is proper, we could use definite integral estimation, like
midpoint rule, Simpson's rule, etc. And we have bounds on how far off
these will be as well.

This is something to consider only when you've thoroughly covered
estimation techniques.

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[Sean Fitzpatrick]

What if we added it as an exploration-type exercise at the end of the section?

I think it's useful to know that we have this control over series for which the integral test applies, since that's so often not the case.

One other note on this section: I agree with how the theorem is stated, where we require that a_n=f(n), where f(x) is a positive, decreasing function.

But the solutions in the examples are written to say things like "let's take the derivative of the function a(n), where we treat a(n) as a continuous function of n".

I think it would be better to say, for example, "Let f(x) = ln(x)/x^2. Then a_n = f(n). Furthermore, f(x)> 0 on (1,\infty), f is continuous by Theorem (xref back to properties of continuous functions), and decreasing, since f'(x) = ..."

[gregory...@gmail.com]

I really like adding the error bounds as an extended exploration exercise. Students can be led to develop the bounds, then apply to a few simple examples.

I must be looking at old versions of the text, as all theorems I see state " Let a sequence {a_n} be defined by a_n=a(n), where a(n) is ...". That is, no reference to a function f. I don't recall if we discussed earlier changing a(n) to f(n). If we already agreed to change a(n) to f(n), then I agree with Sean and we should change the examples to match the theorem.

[Sean Fitzpatrick]

Just pushed an update to the "updates and edits 2021" pull request that has an exercise in it.