The ratio and root tests involve limits at infinity which have a few curious and perhaps counterintuitive properties. The relevant property here is that you can safely ignore/discard/omit any finite number of terms and still have the same valid result. Consider a non-mathematical example: Suppose you had two people running along a track of finite length. Person A starts at some predefined starting point but Person B starts 17 feet further along the track. You can express the positions of the two runners after \(n\) seconds with two sequences \(A_n\) and \(B_n\). The problem we're interested in solving is looking at the limits of the terms of those sequences, as \(n\) approaches infinity.
Now, we could get bogged down with accounting for how fast Person A runs compared to Person B and factoring in Person B's headstart and... but none of that actually matters. The track has finite length, so no matter how fast the two people run, they will eventually reach the end of the track after which point they stop running. That is to say, even though Person B had a headstart on Person A, they ended up at the same point (\(\displaystyle \lim_{n \to \infty} A_n = \lim_{n \to \infty} B_n\)). The analogy gets a bit muddier if we consider a track of infinite length (i.e. sequences with infinitely many terms) but with some thought you should see that the result still holds.
Returning to a mathematical context, the ratio test investigates the limit of the ratio of successive terms of a sequence. Consider the sequence:
\(\displaystyle A = \left\{ 1, 2, 3, 4, 5, \cdots \right\}\)
If we examine the ratios of successive terms we see that the first few ratios are:
\(\displaystyle \frac{2}{1}, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \cdots\)
And:
\(\displaystyle \lim_{n \to \infty} \frac{A_{n+1}}{A_n} = \lim_{n \to \infty} \frac{n+1}{n} = 1\)
Now what would happen if we began looking at the ratio of successive terms but starting at the ninth term instead? The first few ratios would be:
\(\displaystyle \frac{10}{9}, \frac{11}{10}, \frac{12}{11}, \frac{13}{12}, \cdots\)
The limit of these "new" ratios must be exactly the same as the limit of the "old" ratios. Why is that?
Pursuant to your specific examples, it appears to me that the reason the authors have decided to omit just the first term is simply because the first term of the sequence doesn't fit the pattern.
