Does |an|≤Cn−1|an|≤Cn−1,|bn|≤Cn−1|bn|≤Cn−1 means f∈C1(T)f∈C1(T)

Question

I'm trying to proof when |an|≤Cn−1$|a_n|\le C{n}^{-1}$,|bn|≤Cn−1$|b_n|\le C{n}^{-1}$, (|a0|≤C$|a_0|\le C$) Fourier series is converged.
I know when f∈C1(T)$f\in C^1(T)$,Fourier series is converged. but I stuck on the proof of f∈C1(T)$f\in C^1(T)$ (which means f(x) Differentiable)
I doubt conditions are not enough, maybe it also needs |an|≤Cn−2$|a_n|\le C{n}^{-2}$,|bn|≤Cn−2$|b_n|\le C{n}^{-2}$ or a bigger number on the Exponent, then what the number will be?