Sequence Convergence

When a infinite set appears to reach a finite value.

Definition

$\{a_n\}$ converges to $l\in \mathbb{R}$ $⟺$ $\exists l\in \mathbb{R},\forall ϵ>0,\exists N>0$ s.t $\forall n\in \mathbb{N}$ if $n>N$ then, $|a_n-l|<ϵ$

Limit Definition

$\{a_n\}$ converges to $l\in \mathbb{R}$ ${\lim }_{n\to \infty }{a}_n=l$

Cauchy Sequence

Showing Convergence

• Epsilon-Delta Proof
• Convergent properties
• Comparison Theorem for Integrals
• BMCT

Properties of Convergent Sets

• Let $\{a_n\}$, $\{b_n\}$ be sequences
• Let $a,b\in \mathbb{R}$
• If $a_n\to a$
• If $b_n\to b$ Then:

Additive Property

$\{a_n+b_n\}$ converges to $a+b$

Constant Multiple Property

$\forall c\in \mathbb{R},\{c{a}_n\}$ converges to $ca$

Multiplicative Property

$\{a_n{b}_n\}$ converges to $ab$ Converging Sequence Multiplicative Property Proof

Quotient Property

$\left\{\frac{a_n}{b_n}\right\}$ converges to $\frac{a}{b}$ provided $b\ne 0,b_n\ne 0$

Composite Property

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous on $L$ Then, if $\{a_n{\}}_1^{\infty }$ converges to $L$ $\{f(a_1),f(a_2),\dots \}$ converges to $f(L)$

Absolute Property

If ${\lim }_{n\to \infty }|a_n|=0$ Then, $⟹{\lim }_{n\to \infty }{a}_n=0$

Additional Theorems

• Converging Sequence’ Unique Limits
• Monotonic Sequence