1. “Sufficiently large” is an important concept in sequences and series. In essence it means any crazy thing can happen for as long as it wants to happen, so long as there is a finite point after which the sequence or series starts behaving in a controlled/predictable way. A finite number of terms can’t affect the limit, and they have a finite sum and so can only affect the series’ value, not whether it converges or not.

2. Advice I’ve given students: If the series does not look like anything but you’re being asked to evaluate it, try partial fractions and see if you get something telescoping.

3. The limit comparison test asks whether the terms of two series are “proportional in the limit.” The ratio and root tests ask, in two ways, whether the terms of one series are “geometric in the limit.” The geometric series with terms gives ratio r between successive terms (), leading to the ratio test, and the nth root of its nth term is r times the nth root of c (assuming c is positive, and otherwise taking the negation of the series, which has the same convergence behavior), which limits to r, leading to the root test. This is why the cutoff point for convergence and divergence is 1 – that is what it is for geometric series. The distinction, that at limit 1 we don’t know the behavior, is because this is something only “geometric in the limit.”