Solving Finite Series with Real Y

[ahmadmath]

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>

 

[Gerenuk]

Maybe try to express Sine with a complex exponential. Then you get a geometric series. This should work.

 

[ahmadmath]

I had tried but it doesnot work it works just when there are no absolute values

 

[Gerenuk]

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.

 

[ahmadmath]

thanks but I am not sure what you mean

 

[Gerenuk]

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...

 

[ahmadmath]

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

 

[Gerenuk]

At least for x=rational*pi it's fairly easy. But haven't put more thought in it.