Sequence Divergence

A series that reaches positive or negative Infinity.

Proving Divergence

Prove M-N for $\infty$

$\forall M>0,\exists N>0$ s.t $\forall n\in \mathbb{N},n>N⟹a_n>M$

Prove M-N for $-\infty$

$\forall M<0,\exists N>0$ s.t $\forall n\in \mathbb{N},n>N⟹a_n<M$

Prove $ϵ-N$ for Oscillation

$\exists ϵ>0,\forall N\in \mathbb{N},\exists m,n\in \mathbb{N}\text{ s.t }|a_m-a_n|\ge ϵ$

Showing Divergence

• Prove $M-N$ epsilon definition for $\infty$
• Proof divergence by proof of contradiction of Sequence Convergence