Solve this other than punching out actual numbers

  1. [tex] e^{i\pi}=-1 [/tex]

    I was wondering how on earth this was possible. I know that:

    [tex]
    e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!}+...+\frac{z^n}{n!}
    [/tex]

    So

    [tex]
    e^{i\pi}=1+i\pi+\frac{-\pi^2}{2!}+\frac{-\pi^3i}{3!}+\frac{\pi^4}{4!}...
    [/tex]

    I was wondering if there is any way to solve this other than punching out actual numbers and seeing about where they converge to?
     
  2. jcsd
  3. e^ix = cos x + i sin x
     
  4. thanks, I didn't know about that equation
     
  5. mathman

    mathman 6,576
    Science Advisor
    Gold Member

    If you look at the power series for cos(x), sin(x) and eix, the relationship will be obvious.
     
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