The exercise :
I was doing an exercise which consists in finding a necessary and sufficient condition about reals $a_1,a_2, \dots,a_n$ such that $\displaystyle \sum_{k=1}^{n}a_k\text{cotan}\left(kx\right)$ admits a finite limit when $x \rightarrow 0$ and to compute it.
My attempt :
I wrote that $$ \text{cotan}\left(kx\right)\underset{(0)}{=}\frac{1}{kx}-\frac{k}{3}x+o\left(x\right) $$ Hence $$ \sum_{k=1}^{n}a_k\text{cotan}\left(kx\right)\underset{(0)}{=}\sum_{k=1}^{n}\frac{a_k}{k}\frac{1}{x}-\frac{1}{3}\left(\sum_{k=1}^{n}ka_k\right)x+o\left(x\right) $$ Hence a sufficient and necessary condition would be that $\displaystyle \sum_{k=1}^{n}\frac{a_k}{k}=0$ and we would have $$ \sum_{k=1}^{n}a_k\text{cotan}\left(kx\right)\underset{(0)}{=}-\frac{1}{3}\left(\sum_{k=1}^{n}ka_k\right)x+o\left(x\right) $$ And I find that the limit is $0$. I think I'm missing something because the correction of the exercise (that just gives the answer) is the same NSC as me but it found that the limit is $$ L=\frac{1}{3}\sum_{k=1}^{n}a_k $$ Where am I wrong ?
Your limit is correct, theirs is wrong, but you are missing the condition that $$\sum_{k=1}^\infty k a_k$$ converges.