Limit of sequence exists but might be infinity

Question

Assumption 4.2 in Stokey et al. states, for the real sequence $x_t$:

... $\lim_{n\rightarrow \infty} \sum_{t=0}^n x_t$ exists but might be positive or negative infinity.

But this goes against my intuition and understanding.

How can a series going to $\infty$ be converging to a limit? And what is the difference between converging to $\infty$ and diverging?

Source: Stokey, N. & Lucas, R.(1989} Recursive Methods in Economic Dynamics, page 84

Answer 1

This kind of definition is often used and we claim that the limit of a sequences may

• exist finite when $a_n\to L\in \mathbb{R}$ and $a_n$ converges

• exist infinite positive $a_n\to \infty$ and $a_n$ diverges

• exist infinite negative $a_n\to -\infty$ and $a_n$ diverges

• doesn’t exist in all the other cases

Note that for the three cases of existence we need three different definitions.

The advantage of this kind of definition is that we distinguish the infinite cases from the last which is the case of sequences like $\sin n$ for example.

Answer 2

There is no real difference between "diverges to $+\infty$" and "converges to $+\infty$"; the choice of language simply reflects the author's point of view.

When doing calculus/real analysis, it is very convenient to work in the extended real numbers.

$+\infty$ and $-\infty$ are points on the extended real line, and we can talk about limits involving them. we say $\lim_{n \to +\infty} x_n = +\infty$, this is just the ordinary (topological) definition of a limit. And the usual language for limits is that $x_n$ converges to the point $+\infty$ as $n$ goes to $+\infty$.

Introductory calculus classes generally avoid talking about the extended real line. When restricting yourself just to the ordinary real line, such a limit doesn't converge to a point of the real line, so it would be correct to say such a limit does not exist.

These limits, however, are so incredibly useful to know and understand that introductory calculus classes have to teach them, despite never talking about the extended real numbers.

So you have the unfortunate situation where you still want to talk about limits that have infinite values, or whose argument goes to infinity, or both... but since you restrict yourself to a space that does not actually have the points at infinity you can't say these are convergent limits.

In conclusion:

• Saying the "limit exists and has value $+\infty$" means the author is thinking of taking the limits in the extended real numbers
• Saying the "limit does not exist and has value $+\infty$" means the author is thinking of taking limits in the ordinary real numbers, but still finds this situation useful enough to talk about it anyways

Answer 3

"Not converging" can have two meanings

• diverging to $\pm\infty$,

• not converging at all (f.i. with two subsequences that converge to different limits).

The author probably wanted to express compactly that we are not in the second case.