I assume that we are dealing with the PRINCIPAL roots of -1 (when k=0) since for each n there are n
distinct roots of -1 equally spaced about the unit circle. Fistfiz's example using the clock gives a good
illustration of that sequence progressing counterclockwise from e^ipi to 1 around the top of the circle.
or use the composition law for limits to treat it according to this rule:
]]>How does this look? :0)
i*180 i*(180/n) i0
(-1)^(1/n) = (1*e )^(1/n) = 1*e so this approaches 1*e = 1 as n goes to infinity.(The angles are in degrees.)
I have to admit that at first sight this looked funny; but after being (maybe) less superficial i'm seeing a meaning behind this:
look it geometrically (i write polar coordinates for complex numbers)...
the (first) square root for -1 is (1,pi/2) (midnight)
the (first) 3rd root for -1 (1,pi/3) (one o'clock)
the (first) 4th root for -1 is (1,pi/4) (half past one)
.....
..... (...some time passes...)
.....
the (first) nth root for -1 tends to (1,0) (almost three o' clock)
so it seems to me that your limit is what the first nth root of (-1) tends to.
EDIT: I want to add something:
where k=0,1,2...,n-1. In particular, the integer part of (n+1)/2 (which is n/2 if n is even and (n+1)/2 if odd) belongs to the list of k's;(where i put n/2 or n+1/2 as k)
so one of us (or eventually both ) must be wrong.
]]>(The angles are in degrees.)
]]>What do you mean by a succession from N to C?
You see that, for example
]]>While writing my post i realized that if your succession is from N to C it is not even a function, so i don't know if it has any meaning to talk about limit...
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