Consider a sequence $\{a_n\} \to 0$. Does it imply that the series $s_n = \sum\limits_{k = 0}^{n} a_n^2$ converges ?
for example the sequence $a_n = \frac{1}{n}$, $a_n \to 0$ and $\sum\limits_{k=1}^{n}\frac{1}{k^2}$ converges.
Consider a sequence $\{a_n\} \to 0$. Does it imply that the series $s_n = \sum\limits_{k = 0}^{n} a_n^2$ converges ?
for example the sequence $a_n = \frac{1}{n}$, $a_n \to 0$ and $\sum\limits_{k=1}^{n}\frac{1}{k^2}$ converges.
The answer is no. Consider $\{\frac{1}{\ln(n)}\} \rightarrow 0$. But the corresponding sum diverges. You even have that $\ln(n)^2 < n$ for $n$ big enough so it will "diverge faster" than the harmonic series.