In my research, I need to determine unique coefficients $a_k$ in terms $b_k$: $$\sum_{k=0}^n a_k \cos\left(\frac{k}{n+1}t\right)+O\left(t^{2n+1}\right)=\sum_{k=0}^n b_k t^{2k}.$$ This problem showed up in my search of approximations to Riemann $\zeta(s)$ function.
Question 1: Is there a closed form solution for $a_k$?
Question 2: If there is no such closed form solution exist, can we prove the existence of $a_k$?
Question 3: If $n$ goes to infinity, can we prove that $a_k$ exist?