# Limit of a sequence

by KLscilevothma
Tags: limit, sequence
 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|
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P: 5,539
 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.
 P: 321 LOL, thanks Tom and L'hopital lim ln(an+b)/n n->[oo] = lim a/(an+b) n->[oo] =0
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)
 PF Patron Sci Advisor Thanks Emeritus P: 38,429 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.
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?

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