If I wanted to write one plus one, recurring, equals infinity....
Does this make sense?
If so, how would it be written as a formula?
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1$\begingroup$ It does not make sense strictly to speak of anything other than a limit being equal to $\infty$ (and even that, if we are very very strict...). $\endgroup$– MartiganJun 3, 2015 at 11:57
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1$\begingroup$ I thought this was written $\zeta(0)=-\frac12$ or something like that. $\endgroup$– HenryJun 3, 2015 at 13:47
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2$\begingroup$ 4 people think this question is well researched. $\endgroup$– Alec TealJun 3, 2015 at 23:07
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3 Answers
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This depends on how strictly rigorous you want to be. You will, fairly often, see statements such as $$\sum_{i=0}^\infty 1=\infty$$
When this is written the correct interpretation should not really be that the sum of infinitely many $1$'s is infinity. Rather it should be that the limit of partial sums diverges to positive infinity, or written mathematically $$\lim_{n\to\infty}\sum_{i=0}^n1=+\infty$$
There is a very good reason to try to always stay very rigorous when dealing with potential infinites (and even with actual infinities but surprisingly they seem to be less of an issue). That reason is that unless you have a lot of experience and training in mathematics chances are, roughly one out of two statements or ideas you will have about potential infinities, will not only be wrong, but usually won't even make good sense.
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1$\begingroup$ A much better write-up than the Comment I was about to leave! $\endgroup$– hardmathJun 3, 2015 at 12:04
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An alternative way to look at it might be the limit of a recursive function. $$ f(0) = 1 $$ $$ f(n) = f(n-1) + 1 $$ $$ \lim\limits_{n \to \infty} f(n) = \infty $$
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You can say like...
$1 + 1 + 1 ... = \sum\limits_{i=1}^{\infty} 1 = +\infty$
Though this is simply the notation.
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$\begingroup$ It's generally bad practice to make infinity "equal" anything, except in certain contexts. $\endgroup$ Jun 3, 2015 at 14:09
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$\begingroup$ Maybe I didn't make it clear but that's kind of what I was saying when I said "this is simply the notation" - there's a lot more context required for it to be meaningful. $\endgroup$– SubSevnJun 3, 2015 at 14:28
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2$\begingroup$ @user3932000: Of course, those certain contexts are rather pervasive. The aversion you refer to is mainly an artifact of introductory materials being taught in ways that avoid properly introducing $\infty$. $\endgroup$– user14972Jun 3, 2015 at 18:44