Convergence Quiz - True or False

# Convergence Quiz - True or False

 True or False: If the series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} (a_n + b_n)$ are convergent series then $\sum_{n=1}^{\infty} b_n$ may be a divergent series.
 True or False: If the series $\sum_{n=1}^{\infty} \mid a_n \mid$ is divergent then $\sum_{n=1}^{\infty} n \mid a_n \mid$ is also divergent.
 True or False: If the series $\sum_{n=1}^{\infty} \mid a_n \mid^{1/3}$ is divergent then $\sum_{n=1}^{\infty} n^2 a_n$ is also divergent.
 True or False: If the series $\sum_{n=1}^{\infty} a_n^4$ is divergent then $\sum_{n=1}^{\infty} a_n$ is also divergent.
 True or False: If the series $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} a_n \sin a_n$ cannot be conditionally convergent.
 True or False: If the series $\sum_{n=1}^{\infty} a_n$ is convergent then $\sum_{n=1}^{\infty} \frac{1}{n^{\mid a_n \mid}}$ is divergent.
 True or False: If the series $\sum_{n=1}^{\infty} \frac{1}{n^{\mid a_n \mid}}$ is divergent, then $\sum_{n=1}^{\infty} a_n$ is not conditionally convergent.
 True or False: If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are convergent, then $\sum_{n=1}^{\infty} a_nb_n$ is convergent.
 True or False: If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ are convergent, then $\sum_{n=1}^{\infty} a_nb_n$ is convergent.
 True or False: If $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} (a_n + b_n)$ converge conditionally, then $\sum_{n=1}^{\infty} b_n$ may converge absolutely.
 True or False: If $\sum_{n=1}^{\infty} a_n$ is convergent, then $\sum_{n=1}^{\infty} \frac{\cos a_n}{n}$ is convergent.
 True or False: If $a_n$ is divergent, then $\sum_{n=1}^{\infty} n \cos (a_n)$ may be convergent.