Is there a mathematical symbol for "the value grows?"
For example:
This result will be increasingly difficult as the value of n grows to infinity.
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6$\begingroup$ What's wrong with $n \to \infty$? For that matter, what's wrong with "grows to infinity"? $\endgroup$– Qiaochu YuanCommented Oct 2, 2012 at 7:11
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1$\begingroup$ Nothing wrong with the suggestion you proposed, I was just unaware of it's existance/usage. $\endgroup$– Red BananaCommented Oct 2, 2012 at 7:17
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$\begingroup$ If $(x_n)$ is a sequence, then $x_n\uparrow \infty$ means that $x_n$ tends to infinity increasingly, i.e. $x_n\leq x_{n+1}$ and $x_n\to\infty$. But there is really no need to write $n\uparrow \infty$ because $x_n=n$ is obviously increasing. $\endgroup$– Stefan HansenCommented Jan 5, 2013 at 17:23
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$\begingroup$ I've also seen $x_n\nearrow\infty$ for what Stefan writes as $x_n\uparrow\infty$. In general, $x_n\nearrow a$ or $x_n\uparrow a$ would mean that the sequence of the $x_n$ is increasing with limit $a$, and $x_n\searrow a$ or $x_n\downarrow a$ would mean that the sequence is decreasing and has limit $a$. $\endgroup$– Andrés E. CaicedoCommented Aug 3, 2013 at 1:52
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$\begingroup$ Be aware that $x_n\uparrow \infty$ is even the Knuth's up arrow notation for exponentiation i.e. $x_n^\infty$ (whatever this could mean in this situation) $\endgroup$– AlienRemCommented Sep 1, 2015 at 19:05
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2 Answers
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If you were unaware of the symbolic expression $n\to \infty$, I'm wondering whether your audience will appreciate its use. That said, "as $n\to \infty$" means "as the value denoted by $n$ approaches infinity".
You could use an upwards arrow as Stefan suggested above to be more specific that the value is ONLY increasing (not tending toward infinity through ups and downs but strictly increasing), but such upwards arrows are mostly only used by math people (some engineers would likely not even know what it meant). $n\to\infty$ while less specific is much more commonly understood (everyone who remembers their calculus should have no problem reading is as I wrote above).
That said, you were looking for a math symbol, now you've got a choice of two! Have fun!
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"tends to" is different. I prefer the 'equivalency' sign with an arrow beneath. Not only does it convey 'tends to', it specifies 'what' is approaching 'what.' The 'equal' sign with a scroll beneath does not.
