I'm learning about Fourier series, specifically $L^1$ and $L^2$ convergence, and need help with the following exercise:
True or False (justify):
$(1)$ The trigonometric series $2 + \sum_{k = 1}^{\infty}\left(k\cos{kx} + {1 \over k}\sin{kt}\right)$ is the Fourier series of a $2\pi$-periodic function $f\in L^1([-\pi, \pi])$.
$(2)$ The trigonometric series $\sum_{n = 2}^{\infty} {1 \over \log{n}} \cos{nx}$ is the Fourier series of a function $f\in L^2([-\pi, \pi])$.
$(3)$ The trigonometric series $2 + \sum_{k = 1}^{\infty}\left({2 \over \sqrt{k}}\cos{kt} + {1 \over k} \sin{kt}\right)$ is the Fourier series of a function $f\in L^2([-\pi, \pi])$.
Since I'm having difficulties with $(2)$ and $(3)$ I'm going to show my work for $(1)$.
$(1)$ By the Riemann–Lebesgue lemma we know that the Fourier coefficients of $f$ must tend to zero when $k$ approach infinity, i.e. $\lim_{k \to \infty} a_k = \lim_{k \to \infty} b_k =0$. This is certainly true for $b_k = {1 \over k}$ but the limit for $a_k = k$ is clearly divergent as $k \to \infty$. Therefore, the proposition is false.
Is my work correct for $(1)$? I want to apply a similar argument for $(2)$ and $(3)$ but I can't apply the Riemann–Lebesgue lemma since we're looking at convergence in $L^2$.