Register to reply

Limit of a sequence

by KLscilevothma
Tags: limit, sequence
Share this thread:
KLscilevothma
#1
Jun26-03, 05:17 PM
P: 321
It isn't a homework problem but I think I better post it here instead of Mathematics forum, since it belongs to "exam help".

Prove that for any positive real numbers a and b,
lim [(an+b)1/n-1] = 0
n->inf

I don't need to use things like |a-b|<epsilon. A simple way will do. I know it's an easy question but I don't know where to start. Could someone please help.
Phys.Org News Partner Science news on Phys.org
Scientists develop 'electronic nose' for rapid detection of C. diff infection
Why plants in the office make us more productive
Tesla Motors dealing as states play factory poker
Tom Mattson
#2
Jun26-03, 05:43 PM
Emeritus
Sci Advisor
PF Gold
Tom Mattson's Avatar
P: 5,532
Originally posted by KL Kam
Prove that for any positive real numbers a and b,
lim [(an+b)1/n-1] = 0
n->inf
This one just screams "L'Hopital!"

First, rearrange it to:

lim(an+b)1/n=1
n-->&infin;

Then take the natural log of both sides to get:

lim ln(an+b)/n=0
n-->&infin;

This goes to &infin;/&infin;, which is an indeterminate form and ripe for L'Hopital's rule.
KLscilevothma
#3
Jun26-03, 05:58 PM
P: 321
LOL, thanks Tom and L'hopital

lim ln(an+b)/n
n->[oo]

= lim a/(an+b)
n->[oo]
=0

KLscilevothma
#4
Jun27-03, 04:42 AM
P: 321
Limit of a sequence

Oh sorry, I forgot to mention
(an+b)1/n-1
is a sequence, not a function. I think L'hopital's rule applies to differentiable functions only.

Perhaps I better rephase the question a bit.
A sequence {an} is defined by (an+b)1/n-1
Prove that
lim (an+b)1/n-1 = 0
n->inf
(a and b are real numbers and n is a positive integer)
HallsofIvy
#5
Jun27-03, 06:56 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,564
It is true that L'hopital's rule applies to functions rather than sequences.


However, IF we can convert a sequence an to a function f(x) (we can't if the sequence involves things like n! or "floor" or "ceiling" that can't be written simply as a continuous function), then f(x)-> L, an-> L. The other way doesn't necessarily work- the function might not have a limit, depending on how it is defined for non-integer values.
KLscilevothma
#6
Jun27-03, 07:13 AM
P: 321
Originally posted by HallsofIvy
However, IF we can convert a sequence an to a function f(x) (we can't if the sequence involves things like n! or "floor" or "ceiling" that can't be written simply as a continuous function), then f(x)-> L, an-> L. The other way doesn't necessarily work- the function might not have a limit, depending on how it is defined for non-integer values.
So we can treat a sequence as a function if it is an "elementary" one like the one I posted, and can apply L'hopital's rule, is it correct?


Register to reply

Related Discussions
Limit of a sequence Calculus & Beyond Homework 3
Limit of Sequence Calculus & Beyond Homework 6
Limit of sequence equal to limit of function Calculus 1
Limit of a sequence Introductory Physics Homework 3
Limit of a sequence Calculus 28