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| Peter L. Montgomery |
 Posted: Sat Feb 12, 2005 5:06 pm |
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In article <1108240007.618379.45440@g14g2000cwa.googlegroups.com>
"Dave Parker" <dave3852@yahoo.com> writes:
[quote:ae13f36eed]What is an elementary proof of
lim a^n = 0 if |a|<1.
n->oo
that doesn't depend on logarithms or other higher functions?
Consider the case 0 < a < 1. Let x = 1/a - 1. Then x > 0 and[/quote:ae13f36eed]
(1/a)^n = (1 + x)^n >= 1 + nx = 1 + n*(1/a - 1) = (a + n - n*a)/a
Invert to get
0 <= a^n <= a/(a + n - n*a)
You can prove this directly (i.e., without introducing x)
by induction on n when 0 <= a < 1. Use the sandwich theorem.
--
The Weapons of Mass Destruction are tsunamis.
pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA |
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| Dave Parker |
| Posted: Sun Feb 13, 2005 10:02 am |
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Dave Parker wrote:
[quote:6376f1e12a]What is an elementary proof of
lim a^n = 0 if |a|<1
n->oo
[/quote:6376f1e12a]
To everyone who responded: you have answered my question to my
satisfaction, in a variety of interesting and insightful ways. I could
not have asked for a better response, and it exemplifies sci.math at
its
finest. (I'll thank all 5 responders - David Kastrup, Stuart
Newberger,
Johan Mebius, Peter Montgomery, G. A. Edgar - collectively here so as
not to clutter up your busy day with numerous individual posts!)
Dave |
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