Problem 415. Sum the Infinite Series

Created by Roger Stafford

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Last Solution submitted on May 30, 2023

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James on 15 Mar 2012

So what would the Cody size of a brute-force algorithm be for this problem?

David Hill on 7 May 2018

I got the correct answers within 50*eps for all answers except x=0.001 for c. Is c_correct value correct for x=0.001? I put in a small correction factor 1.5e-14 to get your c_correct value.

Rafael S.T. Vieira on 16 Aug 2020

It is possible to obtain the results for this series using Wolfram Alpha or Symbolic Math Toolbox. It is not a pretty result, but the series converges. Unless you are up to hard work, there is no point in finding this simplification manually. And the Taylor Series does not help.

Piotr Balik on 2 Jan 2021

Interesting, getting rough estimate is easy, but without further derivations, being close is hard.
Could this oscillatory behavior be dampened by filtering it for example?

GeeTwo on 15 Jul 2022

I wasted a lot of hours trying to look at these summations as a discrete Fourier transform. (I work in ocean acoustics, so it was natural.) Quick way to get an approximation, however!

Next, I worked out closed forms for multiples of pi/2, but found no enlightenment. And pi/4 multiples got weird, especially on the cosine side.

And contrary to what Rafael S.T. Vieira posted about two years ago, Taylor series was the key to arriving at my solution. I didn't use Wolfram Alpha or Symbolic Toolbox, but I did use the table of integrals in the back of my CRC Handbook of Chemistry and Physics. (Physics major with computer science option (minor) here!). Actually, the integrals I used are also listed in this significantly more modest table: https://www.integral-table.com/.

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Problem 415. Sum the Infinite Series

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