Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o lmer-S\o rensen (MS) protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM using a Fourier series expansion so that the laser amplitude may be represented as a general continuous function. Varying the Fourier coefficients used to generate the pulse produces trade-offs between the laser power, gate time, and fidelity. We specifically study gate-timing errors, and we have shown that the sensitivity of the fidelity to these errors can be improved without a significant increase in the average laser power or the gate time. We plot atomic population vs. time for both the traditional MS protocol and the protocol with AM, highlighting the increased robustness of the AM gates. Our central result is that we improve the leading order dependence on gate timing errors from $\order{\Delta t^2}$ to $\order{\Delta t^6}$, and the protocol allows for arbitrarily high orders of scaling to be achieved in principle.