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I've recently stumbled upon the following problem from Brilliant:

Compute the following:

$$\lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}k$$

Options:

  1. $\displaystyle\int_0^\pi\frac{\cos^2(x)}{x^2}~\mathrm dx$

  2. $\displaystyle\int_0^\pi\frac{\sin^2(x)}{x^2}~\mathrm dx$

  3. $\displaystyle\int_0^\pi\frac{\sin(x)}x~\mathrm dx$

  4. $\displaystyle\int_0^\pi\frac{\cos(x)}x~\mathrm dx$

Already by looking at it, it doesn't make sense for 1 or 4 to be correct, since they both diverge. Instinct then tells me the answer should be 3, since $\sin^2$ doesn't appear in the problem. However, I haven't been able to get to it.

By differentiating, I can find potential maxima as follows:

$$\frac{\mathrm d}{\mathrm dx}\sum_{k=1}^n\frac{\sin(kx)}k=\sum_{k=1}^n\cos(kx)=\csc\left(\frac x2\right)\sin\left(\frac{nx}2\right)\cos\left(\frac{(n+1)x}2\right)$$

Obviously the sum is zero on the boundaries, so we are not interested in them. Aside from them though, there are a lot of points to consider. My suspicion is that the maximum occurs at $x=\frac\pi{n+1}$ since this point has every term in the summand being positive. Furthermore, if this is the case, then we can rearrange the original sum into a Riemann sum, getting 3 as the answer.

How may we continue? Or perhaps this is the wrong approach...?


One may also note that answer choice 2 is less than answer choice 3, and we can see 3 is a possible answer choice, meaning that if this exists and is one of the answer choices, it must be the 3rd one.

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  • $\begingroup$ Which piecewise function has this Fourier series representation for $0 \le x\le \pi$ ? Perhaps this helps. The maximum is around $1.5$ $\endgroup$ – Cesareo Feb 11 '19 at 0:23
  • $\begingroup$ @Cesareo This is a finite sum, and forcing it to be an infinite series doesn't give me any of the answer choices anyways. $\endgroup$ – Simply Beautiful Art Feb 11 '19 at 0:25
  • $\begingroup$ But the limit for $n\to\infty$? $\endgroup$ – Cesareo Feb 11 '19 at 0:26
  • $\begingroup$ Letting $n\to\infty$ and then taking the $\max_{x\in[0,\pi]}$ results in $\frac\pi2$ if I did it right, which does not agree with any of the answers provided. Since the possibility of the maximum being at $\frac\pi{n+1}$ yields a Riemann sum and one of the answers provided strongly implies to me that that is not the correct answer. $\endgroup$ – Simply Beautiful Art Feb 11 '19 at 0:28
  • $\begingroup$ The right result is option $3$ which is equal to $\frac{\pi}{2}+0.89489872236(\frac{\pi}{2}+\frac{\pi}{2})$. The second term is due to the Gibbs phenomenon. $\endgroup$ – Cesareo Feb 11 '19 at 9:22
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The series

$$ S_n(x) = \sum_{k=1}^n \frac{\sin(k x)}{x} $$

represent in the Fourier sense, limiting approximations for the periodic piecewise function

$$ f(x) = \frac{\pi}{2}-\frac x2, \ \ \ 0\le x \lt 2\pi $$

So we have

$$ \lim_{n\to\infty}\left(\max_{0\le x\le \pi}\sum_{k=1}^n\frac{\sin(k x)}{k}\right) = \frac{\pi}{2}+\delta $$

here $\delta$ is due to the Gibbs phenomenon which can be calculated as

$$ \delta = 0.08948987223608363511601442291245487...\times \pi $$

where $\pi$ is the discontinuity jump value.

This result is the same of

$$ \int_0^{\pi}\frac{\sin x}{x} dx = 1.851937051982466... $$

Attached a plot with $n = 10$

enter image description here

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