Tails_of_sequences

Let $\{a_n\}_{n=m}^{\infty}$ be a sequence of reals. Think of $m$ as the starting index of the sequence. Here are some sequences with different starting indices:

For $m=1$:

$$ a_1, a_2, a_3, a_4, a_5, \ldots $$

For $m=3$:

$$ a_3, a_4, a_5, a_6, a_7, a_8, \ldots $$

Now consider a natural number $N \geq m$. A tail of our sequence $\{a_n\}_{n=m}^{\infty}$ is defined as:

$$\{a_n\}_{n=N}^{\infty}$$

Visual Illustration

Let’s take $m=1$ and consider the original sequence starting from index 1:

Original sequence ($m=1$):

$$ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \ldots $$

Now let’s compare this to various tails by choosing different values of $N$:

Tail with $N=1$ (same as original):

$$ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \ldots $$

Tail with $N=4$:

$$ \underbrace{a_1, a_2, a_3}_\text{discarded} \;\Big|\; a_4, a_5, a_6, a_7, a_8, a_9, a_{10}, \ldots $$

Tail with $N=7$:

$$ \underbrace{a_1, a_2, a_3, a_4, a_5, a_6}_\text{discarded} \;\Big|\; a_7, a_8, a_9, a_{10}, \ldots $$

Tail with $N=10$:

$$ \underbrace{a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9}_\text{discarded} \;\Big|\; a_{10}, a_{11}, a_{12}, \ldots $$

As we can see, a tail of the sequence is simply what remains after discarding the first $N-1$ terms. The key insight is that no matter how large we choose $N$, the tail still contains infinitely many terms (we’re just starting the sequence from a later point).