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If $\sum_n a_n$ is convergent, then is $\sum_n a_n^3$ convergent?

If $\sum_n a_n$ is convergent, then is $\sum_n |a_{n+1}^2-a_n^2|$ convergent?

If $\sum_n a_n$ is convergent, then is $\sum_n a_n^2 e^{int}$ convergent?

If $a_n=\frac{j^n}{n^{1/3}}$, then $\sum_n a_n$ is convergent, but $\sum_n a_n^3=\sum_n \frac{1}{n}=+\infty$.

If $a_n = \frac{i^n}{\sqrt{n}}$, then $\sum_n a_n$ is convergent, but $\sum_n |a_{n+1}^2-a_n^2|\geq \sum_n \frac{1}{n}=+\infty$.

If $a_n =\frac{e^{-int/2}}{\sqrt{n}}$, then $\sum_n a_n$ is convergent, but $\sum_n a_n^2 e^{int}=\sum_n \frac{1}{n}=+\infty$.

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