Introduction to Sequences - At A Glance:
No, we aren't talking about dragons or dimes. We can at use a scorpion for comparison, nasty little creatures that they are.
In the proof that a convergent sequence must be bounded, we said that "eventually" the terms had to be close to the limit of the sequence. The idea of "eventually" shows up a lot when talking about sequences and series.
We can split any sequence into two parts at any term am (assume M > 1). The head of the sequence consists of the finitely many terms before am. This is like the head of a scorpion. You care about where his head is at, but only so you can find his tail. The tail of the sequence consists of the infinitely many terms starting at am and continuing forever.
We really only care where the business end of a scorpion, the stinger on his tail, is at. Likewise, when talking about whether a sequence is bounded, we only care about its tail. There are only finitely many terms in the head of the sequence. Those finitely many terms must be bounded, because we can take the biggest and smallest terms to get upper and lower bounds. The only way the sequence can be unbounded is if the terms in the tail zoom off to ±∞.
Just like boundedness, when talking about whether a sequence converges, we only care about its tail. A sequence converges if its terms approach a finite limit L as n approaches ∞. The finitely many terms in the head of the sequence have no bearing on what happens to the terms an as n approaches ∞.
Tails will be very important when we get to series, so it's a good idea to start getting comfortable with them now.