Max of Sum of Sines: Find the Max Value for Even n

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SUMMARY

The discussion centers on determining the maximum value of the function \(\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}\) for even values of \(n\) within the interval \(0 \leq x \leq \pi/2\). It is established that for odd \(n\), the maximum occurs at \(x=0\) with a value of \(\sum_{k=1}^m k^n\). The challenge lies in identifying the maximum for even \(n\) and the conditions under which this occurs, particularly when \(m\) is odd.

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  • Understanding of calculus, specifically differentiation and maxima/minima analysis.
  • Familiarity with trigonometric functions, particularly sine functions.
  • Knowledge of series summation and its properties.
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  • Study the implications of even and odd functions in calculus.
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ekkilop
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Hi!

Consider the function

[itex]\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2[/itex].

If [itex]n[/itex] is odd this function attains its largest value, [itex]\sum_{k=1}^m k^n[/itex] at [itex]x=0[/itex]. But what about if [itex]n[/itex] is even? Where does the maximum occur and what value does it take?

Any help is much appreciated. Thank you!
 
Last edited:
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Please make some attempt at the problem when you ask for help.
i.e. how would you normally go about finding the maximum?
Can you prove the statement about when m is odd?
 

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