Limit of a Sequence

Definition

Convergence of Sequence

${\lim }_{n\to \infty }{a}_n=a\iff \forall ϵ>0,\exists N\in \mathbb{N},\forall n\ge N,|a_n-a|<ϵ$

Suppose $a\in \mathbb{R}$, If $\forall ϵ>0,\exists N\in \mathbb{N},\forall n\ge N,|a_n-a|<ϵ$, then $(a_n)$ is said to converge to the limit $a$

Divergence of Sequence

If $\exists ϵ>0,\forall N\in \mathbb{N},\exists n\ge N,|a_n-a|\ge ϵ$, then $(a_n)$ is said to be divergent

Properties

Arithmetic Operations

If $\underset{n\to \infty }{\lim }{a}_n,\underset{n\to \infty }{\lim }{b}_n$ exists, then the following is satisfied

• $\underset{n\to \infty }{\lim }(a_n\pm b_n)=\underset{n\to \infty }{\lim }{a}_n\pm \underset{n\to \infty }{\lim }{b}_n$
• $\underset{n\to \infty }{\lim }c{a}_n=c\cdot \underset{n\to \infty }{\lim }{a}_n$
• $\underset{n\to \infty }{\lim }(a_n\cdot b_n)=(\underset{n\to \infty }{\lim }{a}_n)\cdot (\underset{n\to \infty }{\lim }{b}_n)$
• $\underset{n\to \infty }{\lim }(\frac{a_n}{b_n})=\frac{\underset{n\to \infty }{\lim }{a}_n}{\underset{n\to \infty }{\lim }{b}_n}$, where $\underset{n\to \infty }{\lim }{b}_n\ne 0$
• $\underset{n\to \infty }{\lim }{a}_n^P=(\underset{n\to \infty }{\lim }{a}_n{)}^P$

Facts

If $(a_n)$ converges, then the limit is unique.

Squeeze Theorem for Sequences

Suppose $(a_n),(c_n)$ be two sequences converging to $L\in \mathbb{R}$, and $(b_n)$ be a Sequence.

If $\underset{n\to \infty }{\lim }{a}_n\le \underset{n\to \infty }{\lim }{b}_n\le \underset{n\to \infty }{\lim }{c}_n$, then $\underset{n\to \infty }{\lim }{b}_n=L$

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