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:

If you interrogate the formula for the dot product, it will gift to you all the explanations you need. You’ve seen mathematicians define the same dot product in different ways.

The Geometric Definition of the Dot Product

In the two-dimensional plane, start with two vectors at the origin. Think of our vectors as line segments that we draw starting at coordinate $(0,0)$ (your origin). The dot product of those two line segments (vectors) equals:

THROW YOUR RULER, an integral will be all you need to measure distance. In fact, the integral can measure distances your ruler cannot.

You’re an engineer tasked with measuring the distance traveled by a rollercoaster on a parabolic path. Let the function $y = x^2$ on the interval $[-1, 1]$ represent that parabola. Keep in mind that the rollercoaster doesn’t exist yet, so we cannot directly go and measure it. We only know that it will follow the shape of the above quadratic.