Looking to get some help with constructing sequence $a_n$ of rational numbers such that $\sum |a_n|$ converges to a rational number but $\sum a_n$ does not.
My thinking is that since one sequence has absolute value then I need to make a sequence that is alternating back and forth.
would the sequence $a_n = (-1)^n$ work? as $\sum |a_n| = 1 $ but $\sum a_n$ would be bouncing back and forth between $1$ and $-1$?
I don't think your example works: $\sum{|(-1)^n|} = |-1| +|1|+|-1|+... = 1+1+1+...$ which diverges. In fact, I don't think $\sum{|a_n|}$ can ever converge if $\sum{a_n}$ doesn't, for the following reasoning: Assume $\sum{a_n}$ diverges to positive infinity; then taking the absolute values of each term can only make the sum larger, i.e. $\sum{|a_n|} > \sum{a_n}$ if for $\sum{a_n} > 0$, and thus $\sum{|a_n|}$ diverges. If it diverges to a negative value, simply negating the sum makes it diverge to a positive value, and taking the absolute value still diverges.