Sequence

pA sequence is an ordered list of real numbers ($a_1,a_2,a_3,\dots$) usually denoted by parenthesis $(a_n)$ or curly brackets $\{a_n\}$.

A sequence can also be rewritten as a function: $f:\mathbb{N}\to \mathbb{R},f(n)=a_n$ and plotted.

Limits of Sequences §

Since sequences only work when n is a natural number, Limit are usually only interesting as $n\to \infty$. The definition of a limit of a sequence is:

${\lim }_{n\to \infty }{a}_n=L$

• If $L$ exists (i.e. is finite, real, etc) then the sequence converges.
• Else, the sequence is said to diverge.

Limit Laws On Sequences §

The same limit laws that work for normal functions work for sequences:

Limit Laws

Just as most operators, like the integral and $\frac{d}{dx}$ have their own respective laws as to what addition, multiplication, etc. mean for them, limits also have certain limit laws. These help decompose a complex limit into a simpler version.

Laws §

1. ${\lim }_{x\to a}[f(x)+g(x)]={\lim }_{x\to a}f(x)+{\lim }_{x\to a}g(x)$

Addition does not affect limits, we can ‘pull out’ the plus sign from the limit. E.g. ${\lim }_{x\to \infty }[x^2+\log (x)]={\lim }_{x\to \infty }(x^2)+{\lim }_{x\to \infty }(\log (x))$

2. ${\lim }_{x\to a}[c\times f(x)]=c\times {\lim }_{x\to a}f(x)$

Constant factors multiplication (and division) do not affect limits. E.g. ${\lim }_{x\to \infty }\left[\frac{39}{x^2}\right]=39\times {\lim }_{x\to \infty }\left(\frac{1}{x^2}\right)$

3. ${\lim }_{x\to a}[f(x)\times g(x)]={\lim }_{x\to a}f(x)\times {\lim }_{x\to a}g(x)$

4. ${\lim }_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}if{\lim }_{x\to a}g(x)\ne 0$

5. ${\lim }_{x\to a}c=cifc\ne \infty$

6. ${\lim }_{x\to a}x=aifa\ne \infty$

Link to source

Continuity On Sequences §

Continuity works basically the same way as it does on a normal function, assuming n is always natural:

Theorem - Continuity & Limits §

If a function can be represented as $f(g(x))$ and ${\lim }_{x\to a}g(x)=b$:

${\lim }_{x\to a}f(g(x))=f({\lim }_{x\to a}g(x))=f(b)$

• $f$ must be continuous at $b$.

This is very, very useful to break down Limit, since we can simplify complex functions into those that are continuous everywhere, and then apply this theorem.

Link to source

Theorem - Sequences & Functions §

Let $f(x)$ be a real-valued function and $(a_n)$ be a sequence of real numbers such that $f(n)=a_n$:

${\lim }_{x\to \infty }f(x)=L⟹{\lim }_{n\to \infty }{a}_n=L$ This theorem means that if a sequence has a valid function that produces the same output values for $n$, then all the same techniques used on Limit of functions (sandwich theorem, L’Hôpital’s rule, etc.) can be applied on sequences.

Not that (as shown via the implies symbol), this does not happen both ways:

${\lim }_{n\to \infty }{a}_n=L/⟹{\lim }_{x\to \infty }f(x)=L$

Limits of sequences do not always hold on functions

Take the sequence $a_n=\sin (2\pi n)$, which always equates zero for $n\in \mathbb{N}$ Hence, the following limit is true for $a_n$ : ${\lim }_{n\to \infty }\sin (2\pi n)=0$

However the same does not apply for the function version: $f(x)=\sin (2\pi x)$ as the value can oscillate between -1 and 1. I.e: ${\lim }_{n\to \infty }\sin (2\pi n)=0/⟹{\lim }_{x\to \infty }\sin (2\pi x)=0$