Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have put some Mathematica code here:

that uses this algorithm:

$$y1 = Sin[x];$$ $$y2 = Sin[y1];$$ $$y3 = Sin[y1 + y2];$$ $$y4 = Sin[y1 + y2 + y3];$$ $$y5 = Sin[y1 + y2 + y3 + y4];$$ $$y6 = Sin[y1 + y2 + y3 + y4 + y5];$$ $$y7 = Sin[y1 + y2 + y3 + y4 + y5 + y6];$$ $$y8 = Sin[y1 + y2 + y3 + y4 + y5 + y6 + y7];$$ $$y9 = Sin[y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8];$$ $$y10 = Sin[y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9];$$ $$y11 = Sin[y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9 + y10];$$ $$y12 = Sin[y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9 + y10 + y11];$$ $$y = y1 + y2 + y3 + y4 + y5 + y6 + y7 + y8 + y9 + y10 + y11 + y12;$$

where $y$ is the purple curve in this image:

enter image description here

The blue curves are $y1$, $y2$, $y3$, $y4$, $y5$, $y6$... and so on.

Does the purple curve $y$ tend to a square wave?

This question builds upon the answer to this previous question.

Edit: The partial sums:

$$y1$$ $$y1 + y2$$ $$y1 + y2 + y3$$ $$y1 + y2 + y3 + y4$$ $$y1 + y2 + y3 + y4 + y5$$ $$y1 + y2 + y3 + y4 + y5 + y6$$ ... and so on, look like this when plotted:

partial sums of iterated sines

For more images see: link to question on dsp stackexchange

share|cite|improve this question
It looks as if it converged pointwise to zero... – Eckhard Jan 12 '13 at 12:58
What does that mean? Pointwise to zero, what is pointwise? – Mats Granvik Jan 12 '13 at 13:02
My comment was referring to the behaviour of $y_n$. The partial sums do indeed seem to converge to a square wave. – Eckhard Jan 12 '13 at 13:42
up vote 2 down vote accepted

The problem is equivalent to analyzing the convergence of the recursive equation

$$z_{n+1}=z_n + \sin(z_n)$$

It's readily seen that the fixed points are $z = k \, \pi$, and that these are attractors only for odd $k$. Further, convergence is guaranteed in each interval (eg, if $z_0\in (0,2\pi)$ then $z_{\infty}=\pi$).

Then, the original sequence $x=\sin(z)$ converges pointwise to zero.

As for its partial sums: the same analysis shows that they are confined to the interval $[0,1]$ in the domain intervals $[0,2 \pi]$, $[4\pi,6 \pi]$,$[8\pi,10 \pi]$ ... , and $[-1,0]$ elsewhere. That's just what gives the dim visual ilussion of a "square wave", if one superposes all partial sums. But it does not converge to a square wave, and each partial sum does not resembles a square wave at all.

share|cite|improve this answer
What do you mean by your comment about the partial sums? Aren't the partial sums just the $z_n$, which as you mentioned do converge towards the square wave? – Johan Feb 19 '13 at 15:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.