Question

Prove that a sequence that diverges to +∞ (resp. −∞) is divergent according

to the previous definition.

Answer 1

Solution:

We say that $\lim _{n \rightarrow \infty} a_{n}=\infty$ if for every number M>0 there is an integer N such that $a_{n}>M \quad \text { whenever } \quad n>N$ .

Next, we say that $\lim _{n \rightarrow \infty} a_{n}=-\infty$ if for every number M<0 there is an integer N such that$a_{n}<M \quad \text{ whenever }\quad n>N$ .

According to the above definition, we have the following:

If $\lim _{n \rightarrow \infty} a_{n}$ doesn't exist or is infinite we say the sequence diverges. Note that we will say the sequence diverges to $\infty$ if $\lim _{n \rightarrow \infty} a_{n}=\infty$ or divergent.

Also, $\lim _{n \rightarrow \infty} a_{n}=-\infty$ we will say that the sequence diverges to $-\infty$ or divergent.