Lecture - Analysis MT22, VIII

Suppose $\exists k \in \mathbb{N} \text{ s.t. } a _ n > 0$ for $n \ge k$. What is true if and only if $\frac{1}{a _ n} \to \infty$ as $n \to \infty$?

\[a_n \to 0\]

Suppose $\exists k \in \mathbb{N} \text{ s.t. } n > k \implies a _ n = -b _ n$ for $n \ge k$. What is true if and only if $b _ n \to \infty$?

\[a_n \to \infty\]

What is the technical definition of $a _ n = O(b _ n)$?

\[\exists c > 0 \text{ and } k \in \mathbb{N} \text{ s.t. } n > k \implies |a_n| \le c|b_n|\]

What’s the technical definition of $a _ n = o(b _ n)$?

\[\exists k \in N \text{ s.t. } n > k \implies b_n \ne 0 \text{ and } \frac{a_n}{b_n} \to 0 \text{ as } n \to \infty\]

What does it mean for a sequence $(a _ n)$ to be monotonic increasing?

\[\forall n > 1: a_n \le a_{n+1}\]

What does it mean for a sequence $(a _ n)$ to be monotonic decreasing?

\[\forall n > 1: a_n \ge a_{n+1}\]

What does it mean for a sequence $(a _ n)$ to be monotonic?

If it is monotonic increasing or monotonic decreasing.

What is the monotonic sequence thoerem about an monotonic increasing sequence $(a _ n)$?

\[(a_n) \text{ is convergent} \iff (a_n) \text{ is bounded above}\]

What key idea is there that you use to prove the monotonic sequence theorem?

Using the completeness axiom.