Understanding notation for the sequence definition

Question

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old?

Definition. A sequence ($s_n$) is said to diverge to $+\infty$ and we write $\lim (s_n) = +\infty$ provided that for every $M$ in $\mathbb R$ there exists a number $N$ such that $n > N$ implies that $s_n > M$. Similarly, $(s_n)$ is said to diverge to $-\infty$ and we write $\lim (s_n) = -\infty$, provided that for every $M$ in $\mathbb R$ there exists a number $N$ such that $n > N$ implies that $s_n < M$.

I start to get confused with the variables as I don't know what some of them mean. For example, why are some capitalized and other not? I know that $\mathbb R$ represents the real number line, and I'm comfortable with its notation; so it seems simple to me that $M$ is just some number on $\mathbb R$.

As I progress through the definition, though, new variables are thrown out with no context (so it seems). For example, it says "$N$ such $n > N$", what are these variables and why is one capitalized one not? Why couldn't two completely different variables be used?

Generally speaking, are there a set a rules to reference when one is reading math definitions (e.g. capital letters typically represent ___,etc)

Any help is greatly appreciated.

Answer 1

Capital N and M in these sort of definitions tend to represent large numbers (including extremely negative). $\epsilon$ and $\delta$ represent numbers near zero.

"For every $M$ in $\mathbb R$ there exists a number $N$ such that $n>N$ implies that $s_n>M$."

For every $M$... that means every. Since it is captialized, think big. No bigger than that.

There is an $N$...If you tell me an $M,$ I will find you an $N.$

Such that $n>N$... $N$ is a floor, and when we are above the floor.

$s_n > M$... our series becomes arbitrarily large.

Now swing it back to the beginning. Every M. Not convinced $s_n$ is unbounded. Choose a higher boundary, and $s_n$ will still bust through it.

Answer 2

Pick any $M$ within the real numbers. Make it as big as you want.

Then a sequence $s_0, s_1, s_2, ...$ diverges to positive infinity if there exists some number $N$ such that $s_{N+1}, s_{N+2}, s_{N+3}, ...$ (in other words, all $s_n$ beyond $s_N$) are all greater than $M$.

This value of $N$ will depend on the particular value of $M$, so it will be in terms of $M$.

Here's an example. Let's define $s_n = 2n$, the sequence of even integers.

For a given $M$ in the real numbers, we can see that $N = \lfloor M/2 \rfloor$ satisfies the definition. Take $M=100$ (even $M$) Then $N=50$, and we observe $s_{51} = 102 > 100$, and also that all following terms are higher. Then take $M=101$ (odd $M$) and do the same thing.

As for notation, each discipline has its own, and each mathematician has his/her own, but it should be carefully defined whatever it is.

Answer 3

Are you familiar with universal quantifiers and existential quantifiers? This might help you a little because it gives us a framework for thinking about these kinds of statements. For example, this statement is:

• $\forall M \in \Bbb{R}$ --> This is the same as saying "choose any $M \in \Bbb{R}$"
• • $\exists N \in \Bbb{N}$ --> This is the same as saying "there is some $N \in \Bbb{N}$". Notice how this goes inside the $\forall M \in \Bbb{R}$ part. This means $N$ is based off of $M$. Someone gives us an $M$, since $M$ can be anything, and we choose a $N$ to make the following true.
• • $n > N \implies s_n > M$ --> This means that for any term beyond the $N^{\text{th}}$ term, the terms are going to be greater than $M$.

Now, we understand that $M$ can be anything and we choose $N$ based off of $M$. If someone gives us $M=1000$, there is some threshold $N$ such that the terms eventually become such that all of them are over $1000$ beyond $N$. If someone gives us $M=10000$, there is some another threshold such that the terms eventually become such that all of them are over $10000$ beyond that threshold.

Basically, what this is saying is that if $s_n$ is going towards $\infty$, then eventually, all of the terms will become over $1000$ and eventually, all of the terms will become over $10000$ and eventually, all of the terms will be over a googol and eventually, all of the terms will over Graham's number, and so on and so on. For any big number $M$, we can find a natural number $N$ such that beyond that $N$, all of the terms will become greater than $M$ and that's what it means to go toward $\infty$: It means that the terms will eventually become bigger than any real number.