Sequences

A 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, limits 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

1. ${\lim }_{x\to a}[f(x)+g(x)]={\lim }_{x\to a}f(x)+{\lim }_{x\to a}g(x)$
2. ${\lim }_{x\to a}[c\times f(x)]=c\times {\lim }_{x\to a}f(x)$
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. ${\lim }_{x\to a}x=aifa\ne \infty$

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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 limits, since we can simplify complex functions into those that are continuous everywhere, and then apply this theorem.

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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 limits 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: ${\lim }_{n\to \infty }\sin (2\pi n)=0/⟹{\lim }_{x\to \infty }\sin (2\pi x)=0$