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

  1. Dec 10, 2015 #1
    I realised 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
     
  2. jcsd
  3. Dec 10, 2015 #2

    BvU

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    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\ \ \ ## :smile:

    --
     
  4. Dec 10, 2015 #3
    Hi BU,
    Thanks for the "cycle". I think you meant ##|{\bf z}| = e## in this case though.
     
  5. Dec 10, 2015 #4

    BvU

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    I certainly do :smile: ! Sorry for the mistake.
     
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