Sy123
This too shall pass
- Joined
- Nov 6, 2011
- Messages
- 3,730
- Gender
- Male
- HSC
- 2013
Help - Real Analysis
So I read somewhere that:
![](https://latex.codecogs.com/png.latex?\bg_white \sum_{k=1}^{\infty} \frac{\sin k \theta}{k} = \frac{\pi}{2} - \frac{\theta}{2} )
However they derived it some weird way that I know nothing about (this is STEP Advanced Problems #43)
Now I set out to derive this, I first though of integrating the sum of![](https://latex.codecogs.com/png.latex?\bg_white \cos k \theta )
And I knew how to derive a closed formula for this sum, via:
![](https://latex.codecogs.com/png.latex?\bg_white z+z^2+z^3+...+z^{n} = \frac{z(1-z^n)}{1-z} )
And substituting z= cis theta in there, and equating real parts, then simplification. All that was easily done, then I had the task of integrating the beast.
I arrived at the result:
![](https://latex.codecogs.com/png.latex?\bg_white \sum_{k=1}^{n} \cos k\theta = \frac{\cos \theta - \cos (n+1)\theta + \cos n\theta -1}{4\cos^2 \frac{\theta}{2} } )
I do realise I need to take a limit for n to infinity, but I will hope to do that later, but I got stuck, I was able to get up to:
![](https://latex.codecogs.com/png.latex?\bg_white \sum_{k=1}^{n} k^{-1}\sin k\theta = \frac{\theta}{2} - \tan \frac{\theta}{2} + \int \frac{\cos n\theta}{4\cos^2 \theta/2} \ d\theta - \int \frac{\cos (n+1) \theta}{4\cos^2 \theta/2} \ d\theta + C )
I plugged in those integrals into Wolfram Alpha and it spat out something horrible so I am assuming I cannot find the exact definite integral for that.
But all I needed to do, was to find:
![](https://latex.codecogs.com/png.latex?\bg_white \lim_{n\to \infty} (I_n - I_{n+1}) )
Where,![](https://latex.codecogs.com/png.latex?\bg_white I_n= \int \frac{\cos n\theta}{cos^2 \theta /2} )
Question 1: The limit is all I need to find in order to complete this yes?
Now, I do know the IBP formula and I have done it just a couple of times, so I barely know anything from it, because I haven't done 4U Integration at school yet. But I have yet to be able to complete this problem, the final goal of it is to be able to show that:
![](https://latex.codecogs.com/png.latex?\bg_white \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + ... )
By utilising a substitution
Question 2: Can anyone give me guidance on how to do this using HSC techniques?
So I read somewhere that:
However they derived it some weird way that I know nothing about (this is STEP Advanced Problems #43)
Now I set out to derive this, I first though of integrating the sum of
And I knew how to derive a closed formula for this sum, via:
And substituting z= cis theta in there, and equating real parts, then simplification. All that was easily done, then I had the task of integrating the beast.
I arrived at the result:
I do realise I need to take a limit for n to infinity, but I will hope to do that later, but I got stuck, I was able to get up to:
I plugged in those integrals into Wolfram Alpha and it spat out something horrible so I am assuming I cannot find the exact definite integral for that.
But all I needed to do, was to find:
Where,
Question 1: The limit is all I need to find in order to complete this yes?
Now, I do know the IBP formula and I have done it just a couple of times, so I barely know anything from it, because I haven't done 4U Integration at school yet. But I have yet to be able to complete this problem, the final goal of it is to be able to show that:
By utilising a substitution
Question 2: Can anyone give me guidance on how to do this using HSC techniques?
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