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;If we accept your and my proceeding then we get:

(where i put n/2 or n+1/2 as k)

so one of us (or eventually both ) must be wrong.

]]>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.)

]]>What do you mean by a succession from N to C?

You see that, for example

]]>here's a short proof of non-existence:

if LIM[a(n)]=L, then for all a(n(k)) LIM[a(n(k))]=L

you see that LIM[a(2k)]!=L since a(2k) is not defined for each k. But maybe someone would argue that for each n in dom(a(n)) a(n)=-1, so LIMa(n)=-1... i see it just as a formal problem, maybe someone can be more precise.

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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