Prove Limit of Sequence: a, b, (an+b)1/n-1 = 0

In summary, the conversation is about proving the limit of a sequence involving positive real numbers a and b. L'Hopital's rule is suggested as a possible method, but it is clarified that the rule only applies to functions rather than sequences. It is mentioned that if the sequence can be converted to an elementary function, then L'Hopital's rule can be used. However, this may not always be possible depending on how the function is defined for non-integer values.
  • #1
KLscilevothma
322
0
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.
 
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  • #2
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.
 
  • #3
LOL, thanks Tom and L'hopital

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

= lim a/(an+b)
n->[oo]
=0
 
  • #4
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)
 
  • #5
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.
 
  • #6
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?
 

1. What is a limit of a sequence?

A limit of a sequence is a value that a sequence approaches and gets arbitrarily close to as the number of terms in the sequence increases. It is denoted as limn→∞ an = L, where L is the limit.

2. How do you prove the limit of a sequence?

To prove the limit of a sequence, we must show that for any positive number ε, there exists a corresponding positive integer N such that for all n ≥ N, the absolute value of (an - L) is less than ε. This can be done using the definition of limit and various limit theorems.

3. What is the significance of the limit of a sequence?

The limit of a sequence helps us understand the behavior of the sequence as the number of terms increases. It can also help us determine the convergence or divergence of a sequence, which has applications in many areas of mathematics and science.

4. What is the formula for proving the limit of a sequence?

The formula for proving the limit of a sequence is limn→∞ an = L, where L is the limit and an is the nth term of the sequence. This formula is based on the definition of limit and can be used to show that a sequence converges to a specific value.

5. Can the limit of a sequence be any number?

No, the limit of a sequence can only be a single value or positive or negative infinity. It cannot be multiple values or complex numbers. Additionally, not all sequences have a limit, as some may diverge to infinity or oscillate between values without approaching a specific value.

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