User:CaptainOdo

For $$ \sum_{k=1}^{\infty} \frac{2+\sqrt{k}}{(k+1)^3-1} $$, use Limit Comparison Test with $$ \sum_{k=1}^{\infty} \frac{1}{k^{\frac{5}{2}}} $$ since the highest term in the numerator is $$ \sqrt{k} $$ and the highest term in the denominator is $$ k^{3}\, $$ and $$ \frac{\sqrt{k}}{k^{3}} = \frac{1}{k^{\frac{5}{2}}} $$.

For $$ \sum_{k=1}^{\infty} \frac{5^{k}+k}{k!+3} $$, use Comparison Test with $$ \sum_{k=1}^{\infty} \frac{5^{k}+k}{k!} $$ and show the convergence/divergence of $$ \sum_{k=1}^{\infty} \frac{5^{k}}{k!} $$ and $$ \sum_{k=1}^{\infty} \frac{k}{k!} $$.