Soit \((a_n)_{n\geqslant 1}\) la suite définie par
\(a_n=(-1)^n\sin\left(\frac{1}{n^{2}}\right)\).
Alors:
- \(\displaystyle \sum_{n=1}^\infty a_n\) converge, mais pas absolument
- \(\displaystyle\sum_{n=1}^\infty (a_n)^2 \) converge, mais
\(\displaystyle \sum_{n=1}^\infty a_n\) diverge
- \(\displaystyle \sum_{n=1}^\infty a_n\) converge absolument
- \(\lim_{n \to \infty} a_n =0\), mais
\(\displaystyle \sum_{n=1}^\infty a_n\) diverge