When you have a formula as in your picture for a sequence, figuring out whether it has a maxima is pretty simple. You can see the equation as a discrete function, with each point representing an element in your sequence. From here, to find whether there is a maxima in your sequence would be the equivalent trying to find whether the function has a maxima or minima. But first, we need to know the interval. Is the sequence defined only for a finite interval? All positive integers? All positive and 0? All negative? All negative and 0? All integers? Depending on the interval, there might or might not be a maxima. In our case, I'll assume that the interval is for all n bigger than -2 since we have a factorial. Now from here, if you want to know if there is a maximum/minimum, you have to figure out if for all n >= -2. This is not an easy task, especially since our function is discrete, but one quick way of doing it is to plot the function, and look at whether any maximas can be seen.
If you were do to that, you would realize that the function swings very quickly at first and after a few points, it converges toward 1, which you could also tell due to the fact that the denominator increases significantly faster than the numerator, leaving us with a small numerator divided by a large denominator which converges toward 0 at n = inf and therefore e^0 = 1. We therefore reach our highest values early on for n = 0 where an = 148.41... and we also end up with a minimum of 0.993 at n = 5. due to the -1^n, we can see that our n2 term keeps switching from positive to negative for even and odd numbers respectively, which affects the value by making the function swing up and down around 1 as n goes to infinity. We therefore end with a function that looks like a decaying sinusoid, which hits it's maxima and minima early on. We can easily find them out by finding the first 5-6 terms and this gives us n = 0 for maxima and n = 5 for minima.