Integrals of limits of sequences of functions

In response to my request to local “non-pure” mathematicians for examples of applications of mathematical analysis in their work, I did receive quite a lot of information relating to interchanging limits and integrals (or limits and series), or differentiating under the integral sign. However, I was thinking about what the most natural examples might be for demonstrating the issues about integrating limits of sequences of functions.

It is quite common to use “moving triangular spike” functions to give examples of sequences of functions which tend pointwise to zero, but whose integrals might do anything. There is nothing wrong with these examples. But perhaps students might think that this is cheating because these functions are not “nice enough”: they don’t expect to meet functions with sharp spikes in very often in real life.

I had a think about some other commonly used sequence of functions on the unit interval, e.g. f_n(x) = x^n and g_n(x) = n x^n (etc.), and these work well on the half-open interval \null [0,1). However the behaviour at x=1, while interesting, may also make the example less convincing to students.

Perhaps the most natural functions to look at on \null [0,1] are functions of the form

h_n(x)= n^\alpha x^n (1-x)

(where \alpha is some constant) because these do converge pointwise to zero on the whole of \null[0,1], but the integrals

\displaystyle \int_0^1 h_n(x) \,{\rm d}x

can be calculated exactly, and the values are asymptotic to n^{\alpha-2}. So, by varying \alpha, you can get examples of every kind you want.

In particular, taking \alpha to be 0, 1, 2, or 3 gives some very natural sequences of polynomials which one can use to illustrate the issues of interchanging limits and integrals.

Let’s take \alpha=2 for the rest of this discussion, so that we are considering the functions h_n(x)=n^2 x^n (1-x).

y=h_10(x) (alpha=2)

Here is a Maple plot of the function h_{10} (when \alpha=2), i.e. the function 100 x^{10} (1-x). Of course, the function doesn’t really take the value zero except at the endpoints (0 and 1) of the interval, but there is a significant proportion of the interval where the function values are very close to zero. As n increases, that region expands, and (for 0\leq x\leq 1) the function values are essentially negligible except on a smaller and smaller interval near x=1. With \alpha=2, as you increase n, the height increases just enough to compensate for this, and the areas tend to a non-zero real constant (1 in fact), as you can check by direct calculation.

n=6,8,10,12,14,alpha=2

Returning to the title of this post, we see here that, for all x \in [0,1], we have \lim_{n\to\infty} h_n(x) = 0, and so

\displaystyle \int_0^1 \lim_{n\to\infty} h_n(x)\,{\rm d}x = 0\,,

but (with \alpha =2) we have

\displaystyle \lim_{n\to\infty} \int_0^1 h_n(x)\,{\rm d}x = 1\,.

Although these  examples may be helpful, I think that it may be even better (following the Cork strategy) to give specific examples of the relevant phenomena that have turned up in the work of colleagues teaching “applied” maths/probability and statistics to the same group of students.

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One response to “Integrals of limits of sequences of functions

  1. I have now added some sketches (Maple plots) to the post above, and some additional commentary.

    Like

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