[kalee20]

For large values of Q, the amplitude of oscillation in an LC circuit hardly decays between one cycle and the next.

The ratio of successive peaks is actually e^(-pi/Q) which is 97% for Q=100; 99.4% for Q=500; etc.

But, we're not looking at the relationship between one cycle and the next; we're looking at the relationship between maximum current and maximum voltage. And these occur, 1/4 cycle apart. So during this time the exponential decay has only kicked in for 1/4 of the time, and this explains why the exponential term in your equation is e^(-pi/4Q).

With so little decay, the error in neglecting the exponential term is, for most purposes, peanuts! So, the Q=100 case gives you, as you say, 0.9921 as a multiplier. So if you neglect it, you get an error of 0.78% in peak voltage, not very much! In fact, the % error is approximately 25 x pi/Q, or just 78/Q percent!