This started innocently enough; I was attempting to explain why we have to be so careful when we attempt to differentiate a power series term by term; that when one talks about infinite sums, the “sum of the derivatives” might fail to exist if the sum is infinite.

Anyone who is familiar with Fourier Series and the square wave understands this well:

yields the “square wave” function (plus zero at the jump discontinuities)

Here I graphed to

Now the resulting function fails to even be continuous. But the resulting function is differentiable except for the points at the jump discontinuities and the derivative is zero for all but a discrete set of points.

(recall: here we have pointwise convergence; to get a differentiable limit, we need other conditions such as uniform convergence together with uniform convergence of the derivatives).

But, just for the heck of it, let’s differentiate term by term and see what we get:

It is easy to see that this result doesn’t even converge to a function of any sort.

Example: let’s see what happens at

And this repeats over and over again; no limit is possible.

Something similar happens for where are relatively prime positive integers.

But something weird is going on with this sum. I plotted the terms with

(and yes, I am using as a type of “envelope function”)

BUT…if one, say, looks at

we really aren’t getting a convergence (even at irrational multiples of ). But SOMETHING is going on!

I decided to plot to

Something is going on, though it isn’t convergence. Note: by accident, I found that the pattern falls apart when I skipped one of the terms.

This is something to think about.

I wonder: for all and we can somehow get close to for given values of by allowing enough terms…but the value of is determined by how many terms we are using (not always the same value of ).