[sdiy] Screwing with Square Waves

[cheater00 .]

Hi Don,

On Sat, Nov 2, 2013 at 6:26 AM, Donald Tillman <don at till.com> wrote:
> On Nov 1, 2013, at 12:05 PM, cheater00 . <cheater00 at gmail.com> wrote:
>
>> On Fri, Nov 1, 2013 at 5:28 PM, Donald Tillman <don at till.com> wrote:
>>> We're summing an infinite number of sines, and each is at a strength of (1/i), inversely proportional to its frequency.  If this was a (1/(i^2)) series, like a Triangle Wave, that converges.  But a (1/i) series does not mathematically converge.
>>
>> A correction has to be made. Obviously the series converges, because
>> there is a limit, and we know it.
>
> We must be talking about different things... The Harmonic Series does not converge.  I was referring to that.  Which series are you referring to?

The harmonic series Sum 1/n does not converge, but it has nothing to
do with a Fourier series with coefficients 1/n. Given this, and the
fact we were talking about harmonic decompositions, I understood you
meant a harmonic series of a function with coefficients 1/n. Those do
converge.

>> More technically, it converges
>> because there is such a function such that the absolute area between
>> that function and a partial sum of our series converges to 0 as the
>> number of terms of the partial series increases:
>>
>> E.f(x) => Int (Sum_{n=0}^k f_n(x) - f(x)) dx -> 0, as k->+oo
>
> I don't know what you mean here.  What exactly is f_n(x)?

I had meant that the harmonic series of a square wave converges. In
the formula above, f_n(x) is a term of the Fourier series of a square
wave. It is 1/(2n+1) sin (2n+1) x. But since you were talking about
Sum 1/n, the above does not apply.

>>
>>> So that infinite energy has to go somewhere.
>>> And as you noted, we see that energy in infinite slopes or infinite spikes.
>>
>> The reference to infinite energy is better put in more technical
>> terms.
>
> I was using the word "energy" in a completely nontechnical way.  Better to refer to it as the contribution from each harmonic.

Sure, but the notion lead to an answer to your question, so it was
good, just needed some TLC.

Cheers,
D.