Limit Comparison Test vs Comparison Test

Sure, use $b_k=\frac{1}{k^{3/2}}$. The limit is of the ratio is $0$, and since $\frac{3}{2}\gt 1$, $\sum_1^\infty \frac{1}{k^{3/2}}$ converges.

There's a different statement of the limit comparison test that would work:

Let $\{{a_n}\}$ and $\{{b_n}\}$ be sequences of nonnegative reals. Then the following hold (the first part is what you can use):

If $\limsup_{k \to \infty}\frac{a_k}{b_k}$ is finite, then convergence of $\{{b_n}\}$ implies convergence of $\{{a_n}\}$.

If $\liminf_{k \to \infty}\frac{a_k}{b_k} > 0$, then divergence of $\{{b_n}\}$ implies divergence of $\{{a_n}\}$.)