PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Low
[parent] limit of nth root of n (Example)

The $ n$th root of $ n$ tends to 1 as $ n$ tends to infinity, i.e. the real number sequence

$\displaystyle \sqrt[1]{1},\, \sqrt[2]{2},\,\sqrt[3]{3},\,\ldots,\,\sqrt[n]{n},\,\ldots$
converges to the limit
$\displaystyle \lim_{n\to\infty}\sqrt[n]{n} = 1.$ (1)

Proof. If we denote $ \sqrt[n]{n} := 1+\delta_n$, we may write by the binomial theorem that

$\displaystyle n = (1+\delta_n)^n = 1+{n\choose1}\delta_n+{n\choose2}\delta_n^2+\ldots+{n\choose n}\delta_n^n.$
This implies, since all terms of the right hand side are positive (when $ n > 1$), that
$\displaystyle n > {n\choose2}\delta_n^2 = \frac{n(n\!-\!1)}{2!}\delta_n^2,\quad \delta_n^2 < \frac{2}{n-1},\quad 0 < \delta_n < \sqrt{\frac{2}{n-1}},$
whence $ \displaystyle\lim_{n\to\infty}\delta_n = 0$. Accordingly,
$\displaystyle \lim_{n\to\infty}\sqrt[n]{n} = \lim_{n\to\infty}(1+\delta_n) = 1,$
Q.E.D.

Note. (1) follows also from the corollary 3 in the entry growth of exponential function.



"limit of nth root of n" is owned by pahio.
(view preamble)

View style:

Other names:  sequence of nth roots of n

This object's parent.

Attachments:
proof of limit of nth root of n (Proof) by rspuzio
Log in to rate this entry.
(view current ratings)

Cross-references: growth of exponential function, positive, side, implies, binomial theorem, limit, converges, sequence, real number, infinity
There is 1 reference to this entry.

This is version 4 of limit of nth root of n, born on 2007-12-07, modified 2007-12-09.
Object id is 10112, canonical name is LimitOfNthRootOfN.
Accessed 691 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 30-00 (Functions of a complex variable :: General reference works )
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
A puzzle. by fishgoldstein on 2008-06-15 15:09:24
Here is a puzzle:

Let's say someone shows you a rod 99 feet in length.
He tells you that you are allowed to cut it up as you see, and you will be given the product of the lengths of all of the pieces in dollars.
For example, if you were to slice it in two pieces 48.5 feet in length, you would be given 48.5 x 48.5 dollars, (2352.25)

How should you cut up the rod?

The answer: make 33 pieces each 3 feet long.
The reason: the nth root of n reaches it's maximum at 3.
[ reply | up ]
Interact
post | correct | update request | add example | add (any)