perplexus.info :: Just Math : Zeta sine

Zeta sine (Posted on 2018-10-28)

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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