Sequences

Definitions 2.0.1  
A sequence is in fact nothing more than an endless list of numbers. There are basically only two kinds of sequences, those who tend to concentrate around one point and those who don't. The following definitions provide a rigorous framework for this intuitive idea.

(a)

A sequence is a map $\hbox {I \hskip -5.2pt {N}}\to \hbox {I \hskip -5.2pt {R}}: n \mapsto x_n$. It is usually denoted by $(x_n)_n$.

(b)

A sequence $(x_n)_n$ is called convergent to a real number $x$ (called the limit point), iff

$$\forall \varepsilon > 0 , \exists n_0 \hbox{ such that } \forall n \geq n_0 : \vert x_n - x \vert < \varepsilon$$ (2.0.1)

This means that for any chosen distance, no matter how small, we have that from a certain point on, all elements of the sequence are closer to the limit point $x$ than that distance. Basically, what happens is that all elements of a sequence ``huddle together" around one single point.
The standard notation for convergence is $x_n \to x$.

(c)

A sequence is called divergent if it is not convergent to some limit point.

Note the similarity of (2.0.1) with (1.2.10), the definition of convergence of a function to a value $L$ for $x \to \infty$. The difference is that in (1.2.10) we are going through a continuum while in (2.0.1) we are going through a discreet set $(x_n)_n$.

Remark 2.0.2  
Note that (2.0.1) is the discrete version of the definition of the limit of a function in $\pm\infty$.

Lemma 2.0.3  
A useful little fact about converging sequence is sometimes referred to as the ``Squeeze Lemma" for sequences: If $x \leq x_n \leq y_n$ for any $n$ and $y_n \to x$, then $x_n \to x$.

Proof.
Indeed, $x_n$ is squeezed all the way to $x$ because we have that $\vert x_n - x\vert \leq \vert y_n - x\vert$ and $\vert y_n - x\vert$ is arbitrarily small, so $\vert x_n - x\vert$ has to become arbitrarily small. $\qedsymbol$

Remark 2.0.4  
The ``Squeeze Lemma" can often save you a lot of tedious calculations when having to calculate a limit point. For example, directly calculating the limit of $x_n = \frac{1}{n} \ln\left(1+\left\vert\sin\left(\frac{n\pi}{8}\right)\right\vert\right)$ can be a very cumbersome task. This is easily avoided, by noticing the following inequality:

$$0 \leq \frac{1}{n} \ln\left(1+\left\vert\sin\left(\frac{n\pi}{8}\right)\right\vert\right) \leq \frac{\ln(2)}{n}$$ (2.0.2)

Then note that $\frac{\ln(2)}{n} \to 0$ and the squeeze lemma gives you $x_n \to 0$.

Examples 2.0.5  
Sequences come in all shapes and sizes. Still, each of the following examples represents a very large class:

(a)

$1,1,1,1,1,1,1,\ldots$ is a constant sequence. Technically it is a sequence, but at the end of the day it might as well be a single number.

(b)

$1,\frac{1}{5},\pi,e,\frac{\pi}{3},7,12,-5,\ldots$ seems to be a sequence without any regularity whatsoever. Therefore, not very interesting, because we don't really know what we are dealing with.

(c)

$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}, \frac{1}{7},\ldots$ is a sequence of positive numbers that seem to approach 0 a little closer every step. Therefore, this is a sequence converging to 0.

(d)

$1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\frac{1}{64},\ldots$ is again, a sequence of which the elements get smaller and smaller. Comparing this sequence to the previous one, this one approaches 0 a lot faster.

(e)

$1,3,5,7,9,11,13,15,17,\ldots$ contains elements that become bigger with every step; in other words, the sequence goes up to infinity. Therefore this sequence is divergent.

(f)

$1,0,1,0,1,0,1,0,\ldots$ has even elements going to 0 and odd elements going to $1$. It is still called a divergent sequence because the elements don't group together around one single point. So, it is divergent. With a somewhat warped terminology, you could say that this sequence diverges to 0 and $1$, as opposed to the previous sequence, which diverges to infinity.

(g)

$1,0,\frac{1}{2},1-\frac{1}{2},\frac{1}{3},1-\frac{1}{3},\frac{1}{4},1-\frac{1}{4},\ldots$ is again a sequence that ``diverges" to 0 and $1$.

Properties 2.0.6  
The following properties of sequences are not very difficult to prove. Still it is worth going through the proofs to see how the `` $\varepsilon$-machinery" works. It all boils down to the same idea: a term of the form $\vert x_n - x\vert$ has to be proven small (i.e. going to 0) using the same property of other terms.

(a)

If $x_n \to x$ and $y_n \to y$, then $\alpha x_n + \beta y_n \to \alpha x + \beta y$ for any real numbers $\alpha$ and $\beta$.

(b)

If $x_n \to x$ and $x_n \to y$, then $x=y$.

(c)

If $x_n \to x$ and $\vert x_n - y_n\vert \to 0$, then $y_n \to x$.

Proof.

(a)

First notice that $\Big\vert (\alpha x_n + \beta y_n) - (\alpha x + \beta y) \Big\vert \leq \vert\alpha\vert \vert x_n-x\vert + \vert\beta\vert \vert y_n-y\vert$. Because $\vert x_n - x\vert$ and $\vert y_n-y\vert$ go to 0, so does the left hand side of the inequality and therefore $\alpha x_n + \beta y_n \to \alpha x + \beta y$.

(b)

We have $\vert x-y\vert \leq \vert x-x_n\vert + \vert x_n-y\vert$, and both $\vert x_n - x\vert$ and $\vert x_n-y\vert$ go to 0, $\vert x-y\vert$ has to be 0. Therefore $x=y$.

(c)

Again, the same idea: $\vert y_n-x\vert \leq \vert y_n-x_n\vert + \vert x_n-x\vert$ and as both terms on the right hand side go to 0, so does $\vert y_n - x\vert$. Therefore $y_n$ converges to $x$.

$\qedsymbol$

Example 2.0.7  
It is not so trivial to see that

$$\lim_{n \to \infty}\frac{a^n}{n!} = 0$$ (2.0.3)

for any positive real number $a$. The first thing to notice is that

$$\frac{a^n}{n!} > \frac{a^{n+1}}{(n+1)!}$$ (2.0.4)

iff

$$1 > \frac{a}{n+1}$$ (2.0.5)

Which means that from a large enough $n$ on, the sequence decreases. Moreover, all terms are positive, so we have a decreasing sequence with a lower bound. Therefore, by the celebrated squeeze lemma, our sequence is convergent. Putting

$$\alpha=\lim_{n \to \infty}\frac{a^n}{n!}$$ (2.0.6)

gives

$$\lim_{n \to \infty}\frac{a^n}{n!}=\lim_{n \to \infty}\frac{a}{n}\cdot\lim_{n \to \infty}\frac{a^{n-1}}{(n-1)!} =0\cdot\alpha =0$$ (2.0.7)

Definitions 2.0.8  

• Euler's number, written ``e" is defined as

  $$e = \lim_{n \to \infty}\left( 1 + \frac{1}{n} \right)^n$$ (2.0.8)

  
  
  A numerical approximation of this number is

  $$e = 2.718281828459045235360287471352662497757247093699959574966967627\ldots$$ (2.0.9)

  
  
  Well, you get the general idea.
• The exponential function is defined as

  $$\exp : \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto e^x$$ (2.0.10)

  
  

• The natural logarithm is then defined as its converse, i.e. the function

  $$\ln : \hbox {I \hskip -5.2pt {R}}\to \hbox {I \hskip -5.2pt {R}}: x \mapsto \ln(x)$$ (2.0.11)

  
  
  is defined by $\ln(\exp(x))=\exp(\ln(x))=x$.

Remark 2.0.9  
The graph of both functions is as follows:

200pt

Note that $\exp(x)$ and $\ln(x)$ are mirrored by the $y=x$ axis, which is how it should be with functions that are each others inverse