The limit of a negative number, raised to the power "n", as n grows

Douweziel

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Hello all, this is my first post! I hope I follow the rules.

I was not sure whether to post this in algebra or calculus but I think Calculus deals with limits more so here we are.

So, what is the limit of a negative number raised to n (like lim(-9/8)n with n->∞)? Since even numbers will yield positive infinity and odd will yield -∞, how is this formulated (if possible)?

Thank you in advance! :)
 
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As you've noted, the expression (-9/8)^n is positive if n an is an even number, and negative otherwise. Accordingly, if infinity were even, we'd expect the limit as n approaches infinity to be positive infinity. On the other hand, if infinity were odd, we'd expect the limit to be negative infinity. However, infinity is not a number in the conventional sense, so it's neither odd nor even. There are extensions of the real numbers that treat ∞ as the number explicitly defined as being greater than all real numbers, and -∞ as the number less than all real numbers, but even there ±∞ are neither real numbers nor complex numbers, meaning they do not obey many of the properties of numbers. As a result, again, it is neither even nor odd.

For the purpose of this limit, the correct terminology is that it is undefined or does not exist. A limit that oscillates (switches) between two or more values is one of the ways in which a limit is said to not exist or be undefined.
 
So, what is the limit of a negative number squared (like lim(-9/8)n with n->∞)? Since even numbers will yield positive infinity and odd will yield -∞, how is this formulated (if possible)?
You refer to "a negative number squared", but then show the negative number being raised to an arbitrary natural number. Where, specifically, did you mean to include the squaring? Or, by "square", did you mean "arbitrary whole-number value"?

Thank you! ;)
 
You refer to "a negative number squared", but then show the negative number being raised to an arbitrary natural number. Where, specifically, did you mean to include the squaring? Or, by "square", did you mean "arbitrary whole-number value"?

Thank you! ;)
Mixup - I'm Dutch! I meant "to the power of n" - will change it. Thank you for pointing out this quite important difference.
 
As you've noted, the expression (-9/8)^n is positive if n an is an even number, and negative otherwise. Accordingly, if infinity were even, we'd expect the limit as n approaches infinity to be positive infinity. On the other hand, if infinity were odd, we'd expect the limit to be negative infinity. However, infinity is not a number in the conventional sense, so it's neither odd nor even.

Didn't think of it that way, thank you for clarifying :)
 
Don't seem to be able to change the title of my own thread - can someone else change this for me? Thanks in advance!
 
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