Showing that nth root of c_n is equal to nth root of c_n+1 in the limit

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In summary, the conversation discusses the attempt to prove that $\lim \sup \sqrt[n]{c_{n+1}}=\lim \sup \sqrt[n]{c_n}$ and the use of exponential properties to show this. The possibility of $c_n$ being equal to 1 for all $n$ is also mentioned.
  • #1
OhMyMarkov
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Hello everyone!

I'm trying to show that $\lim \sup \sqrt[n]{c_{n+1}}=\lim \sup \sqrt[n]{c_n}$

This is my attempt:
$\lim \sup \sqrt[n]{c_{n+1}} = \lim \sup \sqrt[m-1]{c_m}=\lim \sup c_m \; ^{\frac{1}{m}}c_m \; ^{\frac{1}{m(m-1)}}$

I'm stuck here, I think I must use some exponential property that says that something decays faster than something or the ratio of two things is zero in the limit...

Any help is appreciated!
 
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  • #2
OhMyMarkov said:
I'm trying to show that $\lim \sup \sqrt[n]{c_{n+1}}=\lim \sup \sqrt[n]{c_n}{n}$
What if $c_n=1$ for all $n$?
 
  • #3
Ah excuse me LaTeX typo: I meant

I'm trying to show that: $\lim \sup \sqrt[n]{c_{n+1}} = \lim \sup \sqrt[n]{c_{n}}$

I fixed it in the thread
 

FAQ: Showing that nth root of c_n is equal to nth root of c_n+1 in the limit

What does it mean to show that the nth root of c_n is equal to the nth root of c_n+1 in the limit?

Showing that the nth root of c_n is equal to the nth root of c_n+1 in the limit means to prove that as n approaches infinity, the values of c_n and c_n+1 converge to the same value when taking their nth root. In other words, the difference between the nth roots of c_n and c_n+1 becomes negligible as n gets larger.

Why is it important to establish the equality of these nth roots in the limit?

This equality is important because it allows us to simplify complex mathematical expressions and make them easier to work with. It also helps us understand the behavior of the sequence of values as n approaches infinity.

How can one prove that the nth root of c_n is equal to the nth root of c_n+1 in the limit?

To prove this, one can use various mathematical techniques such as the squeeze theorem, the limit comparison test, or the ratio test. These methods involve manipulating the given expression and applying known rules and properties of limits to show that the two nth roots converge to the same value as n approaches infinity.

Are there any assumptions or conditions that need to be met for this equality to hold?

Yes, in order for the equality of the nth roots to hold in the limit, the sequence of values c_n must be convergent and the limit must exist. Additionally, the values of c_n and c_n+1 should not be equal to 0 at any point in the sequence, as this would result in an indeterminate form.

What are some real-life applications of this concept?

This concept is commonly used in calculus, particularly in the study of limits and sequences. It also has applications in various fields of science, such as physics, engineering, and finance, where we often encounter sequences of values that tend towards a limit. Additionally, understanding this concept is important in the development and analysis of algorithms in computer science.

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