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This is not an answer but it is too long for a comment. Using a CAS, I found that $$S_m=\sum_{n=1}^m \frac{e^{2 i \pi n \left(1+\frac{1}{2 m+2}\right) }}{n}=-\left(e^{\frac{i \pi }{m+1}}\right)^{m+1} \Phi \left(e^{\frac{i \pi }{m+1}},1,m+1\right)-\log \left(1-e^{\frac{i \pi }{m+1}}\right)$$ where appears the Lerch transcendent function. However, I ...


$G$ is a constant. It's one number, not a variable ranging over numbers. You don't "treat it as a variable", and it's probably misleading or confusing (to yourself) to "assume that it's a varlable". Similarly, you can't "take the derivative of a constant with respect to itself" — it's borderline nonsense to say that or try to think it. But you can treat ...


Hint You need to have continuity at the points where the function's definition is changing. So you need \begin{align*} f(-1) & = \lim_{x \to -1}f(x)\\ f(1/2) & = \lim_{x \to 1/2}f(x). \end{align*} Now solve for $a,b$.

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