Can a sequence be undefined at a point?

Question

A sequence which is a mapping from $\mathbb{N} \to \mathbb{R}$.

For example can the sequence $\{a_n\} = 1/(3-n)$. This would be undefined at $3$. Is it a sequence?

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score 8 · Answer 1 · answered Nov 17 '18 at 08:43

8

If you define a sequence as a mapping $f$ from $\Bbb N$ to $\Bbb R$, then no, a sequence cannot be undefined at a point $x$, since if $f(x)$ was not defined, then $f$ isn't a mapping.

answered Nov 17 '18 at 08:43

Joey Kilpatrick

1,515

2

+1 for "just the facts", which is about all one can do unless the OP provides more information or context for the question. I was trying to write a comment about how $\frac{1}{3-n}$ certainly defines a sequence if we begin with $n=4$ or later, and how the same infinite list of numbers can have different functions showing they're sequences, then started to trip over too many things, after which I pretty much decided to give it a pass, and then your answer showed up. – Dave L. Renfro Nov 17 '18 at 08:47

score 2 · Answer 2 · answered Nov 17 '18 at 10:33

If it's a sequence of real numbers, then go ahead and start noting the first few terms. In your case it will be:

$\frac{1}{2}, 1, \infty,-1, \frac{-1}{2},...$

Is that really a sequence of real numbers? I'd say no, because $\infty$ is not a real number.

Can a sequence be undefined at a point?

2 Answers2