If $\sum{a_n}$ converges does that imply that $\sum{\frac{a_n}{n}}$ converges?

You might want to consider Dirichlet's test.

The problem can be handled by a summation by parts. Define $s_N:=\sum_{j=1}^N a_j$. Then for $M<N$, \begin{align} \sum_{j=M}^N\frac{a_j}j&=\sum_{j=M}^N\frac{s_j-s_{j-1}}j\\ &=\sum_{k=M}^N\frac{s_k}k-\sum_{k=M-1}^{N-1}\frac{s_k}{k+1}\\ &=\frac{S_N}N-\frac{s_{M-1}}M+\sum_{k=M}^{N-1}\frac{s_k}{k(k+1)}, \end{align} which yields the bound $$\left|\sum_{j=M}^N\frac{a_j}j\right|\leqslant \sup_k|s_k|\left(\frac 1M+\frac 1N+\sum_{k\geqslant M}\left(\frac 1k-\frac 1{k+1}\right)\right)\\ \\=\sup_k|s_k|\left(\frac 2M+\frac 1N\right).$$ This proves that the sequence $\left(\sum_{j=1}^Na_j/j\right)_{N\geqslant 1}$ is Cauchy, hence the convergence of the series $\sum_{j\geqslant 1}a_j/j$.