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It seems to me that if we introduce such numbers, due to Fourier series we would be able to express the value of any periodic function at infinity as a trigonometric series of the above values.

It would be also possible to express the values of functions such as $\cos 1/x$ at zero, making them continuous.

For instance, if we define $w=(-1)^\infty$ then $\cos^2(\infty)=(w+\frac1w)/2+1/2$

We also would possibly be able to assign precise values to the series like $\sum_{k=0}^\infty (-1)^k$

Bill Dubuque
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Anixx
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    Sometimes it happens that putting together a bunch of things that don't make sense separately, you can get a consistent new interesting theory. I doubt that this is the case here. – Arnaud Mortier Dec 21 '19 at 15:52
  • See also this question. – Dietrich Burde Dec 21 '19 at 15:58
  • @Dietrich Burde this is a question about algebra, I know what is indeterminate form! – Anixx Dec 21 '19 at 16:02
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    The answers there say that indeed people tried to introduce $\sin(\infty)$ as numbers, however ....I think there are many helpful answers which go much further than "indeterminate form" and also answer this question. – Dietrich Burde Dec 21 '19 at 16:04
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    Why would being able to express the value of periodic functions at infinity or having $y=\cos\frac1x$ being continuous at $0$ be useful to mathematicians? This seems like a solution in search of a problem. –  Dec 21 '19 at 16:48
  • @Matthew Daly well I do not claim it would be useful, I just wonder what would happen and what algebraic properties such a number would have. – Anixx Dec 21 '19 at 17:02

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$(-1)^n$ alternates between 1 and -1 and so thinking about $(-1)^{\infty}$ doesn't really make sense. Similarly sine and cosine are oscillating waves that do not tend towards a fixed value in the real line. A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. The sum is infinite, not the arguments of the cosine and sine functions.

Norse
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Alessio K
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  • I was talking not about assigning a real value but a special value that is not a real number. – Anixx Dec 21 '19 at 16:13
  • There's is no "symbol" that I'm aware of that represents these values, but the point is that these values are undetermined. – Alessio K Dec 21 '19 at 16:14
  • The 1 - 1 + 1 - 1 + 1 - 1 .. series also diverges, alternates b/w 0 and 1 but we have assigned a value of +1/2 to it. It seems like this should be possible. – zombiesauce May 21 '20 at 15:31