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Solve this other than punching out actual numbers

by kreil
Tags: actual, numbers, punching, series, solve
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kreil
#1
Nov21-04, 08:15 AM
kreil's Avatar
P: 545
[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?
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CTS
#2
Nov21-04, 10:59 AM
P: 20
e^ix = cos x + i sin x
kreil
#3
Nov21-04, 11:22 AM
kreil's Avatar
P: 545
thanks, I didn't know about that equation

mathman
#4
Nov21-04, 03:50 PM
Sci Advisor
P: 6,068
Solve this other than punching out actual numbers

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


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