Take a sequence of non-negative numbers $(t_k)_{k=1}^\infty$ with $\lim\limits_{k\to\infty}t_k=0$ (the sequence is not necessarily monotonously decreasing). I'm interested in the relationship between the following two conditions:
$$ \sum_{k=1}^\infty \frac{t_k}{k}=\infty \tag{1} $$
$$ \sum_{k=1}^\infty t_k^2 = \infty. \tag{2} $$
The first condition implies that $(t_k)$ converges to zero slower than $k^{-\alpha},$ for any $\alpha>0.$ Otherwise the series on the LHS of $(1)$ would converge by comparison with the Riemann-Zeta function.
Similarly, the second condition implies that $(t_k)$ converges to zero slower than any $k^{-1/2-\alpha},$ for any $\alpha>0.$
For example, the sequence $(1/\sqrt{k})_{k=1}^\infty$ satisfies the second condition (Harmonic series) but not the first one (Riemann-Zeta function).
The example shows that the second condition does not imply the first condition.
Question: Does the first condition imply the second condition?
Hint: Suppose $\sum t_k^2 < \infty.$ Apply Cauchy-Schwartz to the other series.