Question

What is a divergent infinite series?

Answer 1

The sum `S` of a series `sum_{k=1}^infty a_k` is defined as
`S=lim_{n to infty}S_n`,
where `S_n` is the nth partial sum defined as
`S_n=a_1+a_2+a_3+cdots+a_n=sum_{k=1}^na_k`.

So, a series is called divergent when the limit (the sum)

`S=lim_{n to infty}S_n`

does not exist (including infinite limits).