Problem 415. Sum the Infinite Series
Given that 0 < x and x < 2*pi where x is in radians, write a function
[c,s] = infinite_series(x);
that returns with the sums of the two infinite series
c = cos(2*x)/1/2 + cos(3*x)/2/3 + cos(4*x)/3/4 + ... + cos((n+1)*x)/n/(n+1) + ... s = sin(2*x)/1/2 + sin(3*x)/2/3 + sin(4*x)/3/4 + ... + sin((n+1)*x)/n/(n+1) + ...
Solution Stats
Problem Comments
-
5 Comments
So what would the Cody size of a brute-force algorithm be for this problem?
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.
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.
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?
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/.
Solution Comments
Show commentsProblem Recent Solvers128
Suggested Problems
-
2407 Solvers
-
Calculate the area of a triangle between three points
2919 Solvers
-
509 Solvers
-
5029 Solvers
-
Mersenne Primes vs. All Primes
600 Solvers
More from this Author3
Problem Tags
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!