Discuss whether or not it is possible to have a Fourier series $$a_0+\sum_{k=1}^\infty[a_k\cos(kx)+b_k\sin(kx)]$$ converge for all $x$ without either $$a_0+\sum_{k=1}^\infty a_k\cos(kx) \text{ or } \sum_{k=1}^\infty b_k\sin(kx)$$ converging.
This is a problem in Bressoud's analysis book and my solution is as follows: "No, because if we let $f(x)=a_0+\sum_{k=1}^\infty[a_k\cos(kx)+b_k\sin(kx)]$, then the two other series are obtained by taking $\frac{f(x)\pm f(-x)}{2}$ and since $f(x)$ is convergent for all $x$ the sine and cosine series should also be convergent."
Here is the hint from the back of the book:
If the Fourier series converges at $x=0$, then $\sum_{k=1}^\infty a_k$ converges, and therefore the partial sums of $\sum_{k=1}^\infty a_k$ are bounded.
Although I think my solution is correct (please correct me if I'm wrong) I still would like to see other solutions and in particular understand the author's hint since I can't see how the boundedness of partial sums can help.
Thanks!