Ratio test : first of all, if $\displaystyle \frac{a_{k+1}}{a_k} \rightarrow \infty$, then the sequence does not converge, which means the series $\displaystyle \sum a_k$ surely won't converge.
Same with the root test - if limsup$\displaystyle (a_k)^\frac{1}{k} \rightarrow \infty$, then $\displaystyle a_k \rightarrow \infty$, and the sequence diverges, so the series will diverge too.