The question says that: Sum the series
I have solved the answer as follows:
As the above picture, I don’t know what should I do after the step. The question asks to solve the problem using geometric progression or binomial theorem for complex quantities. Please help.
Hint
You properly arrived to $$C+iS=1+\sum_{k=1}^n (-1)^k (k+1) e^{i a k}$$ Multiply both side by $e^{ia}$ to get $$(C+iS)e^{ia}=\sum_{k=0}^n (-1)^k (k+1) e^{i a (k+1)}$$ Integrate the rhs and let $x=e^{ia}$ to face something you know.
When done, differentiate the result.