Conditions that satisfy Fourier in a particular situation

Question

Proposition: Complex sequence an$a_n$,bn$b_n$(n$n$ from 0$0$ to ∞$\infty$),For a number C>0$C>0$,
|a0|≤C$|a_0|\le C$, |an|≤Cn−k$|a_n|\le C{n}^{-k}$,(n≥1)$(n\ge 1)$,|bn|≤Cn−k$|b_n|\le C{n}^{-k}$,(n≥1)$(n\ge 1)$
If it meets, the series

a02+∑i=1∞(ancosnx+bnsinnx)$$\frac{a_0}{2}+\sum _{i=1}^{\infty }(a_{n}cosnx+b_{n}sinnx)$$

Converges for all x∈R$x\in R$, and its limit as a function of x, is continuous as a whole.
I want to find the minimum value of k(a natural number) for which the proposition holds.
Can anyone help me?