Multivalued limit of (i + i/n)^n

[forcefield]

I realized that (i + i/n)ⁿ approaches 4 discrete values (e, ei, -e and -ei) as n approaches infinity (if n is integer). (If I take that i² = 1, then it approaches two discrete values (e and ei)). Does this kind of "multivalued limit" have some other name so I can learn more about it or where it has been applied ?

Why is eⁱ not equal to this limit (since e¹ is equal to the limit (1 + 1/n)ⁿ as n approaches inifinity) ? I know it's a matter of definitions but why isn't it defined this way ?

Thanks

 

[BvU]

Hi ff,

"approaches 4 discrete values" is a little bit self-contradictory. Perhaps it's more sensible in this case to replace n by x (x a real number) and speak of a limit cycle (the circle $|{\bf z}| = 1$) in the complex plane

"Why is eⁱ not equal to this limit (since e¹ is equal to the limit (1 + 1/n)ⁿ as n approaches inifinity) ?"

That is because $e^i = \lim \,({\bf 1}+i/n)^n\ \ \$

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[forcefield]

Hi BU,

Perhaps it's more sensible in this case to replace n by x (x a real number) and speak of a limit cycle (the circle $|{\bf z}| = 1$) in the complex plane

Thanks for the "cycle". I think you meant $|{\bf z}| = e$ in this case though.

 

[BvU]

I certainly do ! Sorry for the mistake.