If a trigonometric series converges to a function $f$, is the Fourier series of $f$ the original series?

Question

If a trigonometric series $\frac{1}{2}a_0+\sum_{n=1}^\infty (a_n\cos(nx)+b_n\sin(nx))$ converges pointwise to a function $f$, then is the Fourier series of $f$ the original trigonometric series?

I know that if we have uniform convergence, then we can integrate term by term and the conclusion is true. But what if we only have pointwise convergence?

I appreciate any help!