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Zeta sine (Posted on 2018-10-28) Difficulty: 4 of 5
What is the maximum value of the expression below?

sin x / 13 + sin 2x / 23 + sin 3x / 33 + ...

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution (spoiler) | Comment 1 of 6
Looks like a candidate for the Squeeze Theorem.

For all real x and k:   -1/x^3   <=  sin kx / x^3   <=   +1/x^3  
(because -1 <= sin x  <= 1 for all real x)

Hence, using SS to mean Sigma {1 ... N} 

==>     SS -1/k^3  <=  SS sin kx/x^3  <=   SS 1/x^3 

Then, in the limit as N --> infinity:
  -0.120205... <=  SS {infinity} sin kx/x^3   <=  0.120205... (Apery's constant)

So, maximum value = 0.120205... 

  Posted by JayDeeKay on 2018-10-31 11:08:07
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