Question 9
Decide, with proof, if each of the following statement is true.
a. If $(a_n)_{n \ge 1}$ is a decreasing sequence with positive terms such that $\sum a_n = +\infty$ then there exists a decreasing sequence $(b_n)_{n \ge 1}$ with positive terms with $b_n \le a_n$ for all $n$, $\sum b_n = +\infty$ and $\lim(n b_n) = 0$.
b. If $(a_n)_{n \ge 1}$ is a decreasing sequence with positive terms such that $\sum a_n = +\infty$ then there exists a decreasing sequence $(b_n)_{n \ge 1}$ with positive terms with $b_n \le a_n$ for all $n$, $\sum b_n = +\infty$ and $(n b_n)_{n \ge 1}$ decreasing.
a. If $(a_n)_{n \ge 1}$ is a decreasing sequence with positive terms such that $\sum a_n = +\infty$ then there exists a decreasing sequence $(b_n)_{n \ge 1}$ with positive terms with $b_n \le a_n$ for all $n$, $\sum b_n = +\infty$ and $\lim(n b_n) = 0$.
b. If $(a_n)_{n \ge 1}$ is a decreasing sequence with positive terms such that $\sum a_n = +\infty$ then there exists a decreasing sequence $(b_n)_{n \ge 1}$ with positive terms with $b_n \le a_n$ for all $n$, $\sum b_n = +\infty$ and $(n b_n)_{n \ge 1}$ decreasing.