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#1 2012-11-05 21:45:47

mitu
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limit

what is the limit of (-1)^(1/n) when n tends to infinity? does it exist?

#2 2012-11-06 02:05:34

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,859

Re: limit

I would say it is either 1 or non-existent...


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2012-11-06 04:55:47

Fistfiz
Member
Registered: 2012-07-20
Posts: 33

Re: limit

hi mitu, I would say it does not exist if your succession is from N to R;
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...


30+2=28 (Mom's identity)

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#4 2012-11-06 05:00:00

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,859

Re: limit

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2012-11-06 07:49:10

Fistfiz
Member
Registered: 2012-07-20
Posts: 33

Re: limit

anonimnystefy wrote:

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


You see that, for example

Last edited by Fistfiz (2012-11-06 07:51:38)


30+2=28 (Mom's identity)

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#6 2012-11-06 08:05:47

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: limit

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.)


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#7 2012-11-06 08:28:14

Fistfiz
Member
Registered: 2012-07-20
Posts: 33

Re: limit

noelevans wrote:

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 dizzy) must be wrong.

Last edited by Fistfiz (2012-11-06 09:00:00)


30+2=28 (Mom's identity)

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#8 2012-11-06 09:51:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 82,549

Re: limit

Hi;

or use the composition law for limits to treat it according to this rule:


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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#9 2012-11-06 18:35:49

noelevans
Member
Registered: 2012-07-20
Posts: 236

Re: limit

Hi!

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.

smile


Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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#10 2012-11-06 19:54:34

princess snowwhite
Member
Registered: 2012-11-06
Posts: 29

Re: limit

Thank you all. You have been most helpful. Mitu

Last edited by princess snowwhite (2012-11-06 19:55:03)

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