Solving Finite Series with Real Y

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    Finite Series
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Discussion Overview

The discussion revolves around finding the sum of a finite series involving the absolute values of sine functions, specifically the series |ysin(x)|+|y2sin(2x)|+...+|ynsin(nx)|, where y is a real number. Participants explore various approaches to tackle the problem, including the use of complex exponentials and logical reasoning to handle the absolute values.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests expressing sine using complex exponentials to transform the series into a geometric series.
  • Another participant notes that the approach with complex exponentials fails when absolute values are involved.
  • A different perspective emphasizes the need to analyze the signs of the terms before applying the modulus, proposing to sum positive and negative terms separately.
  • One participant expresses uncertainty about the feasibility of predicting which terms will be positive or negative.
  • Another participant believes that for rational multiples of pi, determining the signs of the terms becomes more manageable.

Areas of Agreement / Disagreement

Participants express differing views on the difficulty of predicting the signs of terms in the series and whether certain approaches can lead to a solution. There is no consensus on a definitive method to solve the problem.

Contextual Notes

Participants acknowledge the complexity introduced by the absolute values and the dependence on the value of x, particularly when x is a rational multiple of pi. There are unresolved calculations and assumptions regarding the behavior of the sine function in this context.

ahmadmath
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I cannot figure out the sum of this finite series:
|ysin(x)|+|y2sin(2x)|+...+|ynsin(nx)|
where y is real.
so I want to listen any opinion may help me>
 
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Maybe try to express Sine with a complex exponential. Then you get a geometric series. This should work.
 
I had tried but it doesnot work it works just when there are no absolute values
 
Be creative ;) It works if you work through the cases, i.e. use logic to split positive and negative terms. It's messy, but gives you a closed form equation in the end.
 
thanks but I am not sure what you mean
 
I realize it's harder than I thought first.
I meant first determine which terms are going to be negative and which ones are positive (before the modulus).
Then sum both sets separately, because knowing the sign you can actually remove the modulus and replace it by *(-1) wherever you determined the sign to be negative. Depending on x you will have some runs of positive only terms and some negative only terms.
If x is a rational part of pi then it should be predictable. I haven't completely the calculation though...
 
I think it is so hard to do that in this manner because it is so hard to predict which terms should be positive or negative
 
At least for x=rational*pi it's fairly easy. But haven't put more thought in it.
 

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