Proving Unboundedness of {xn} with [(n+1)/n]^3 - n^3

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In summary, the task is to show that the sequence {xn} = [(n+1)/n]^3 - n^3 is unbounded. One approach is to consider the function f(x) = [(x+1)/x]^3 - x^3 and show that it is monotonically decreasing for all x>0 and has a range of (0,infinity). This can then be used to conclude that f(n) is also unbounded. Additional clarification and guidance is provided in the conversation.
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Homework Statement



show that the following sequence is unbounded

{xn}= [(n+1)/n]^3 - n^3

Homework Equations





The Attempt at a Solution



I expanded it but didnt get anywhere after that. I know that a sequence is unbounded if for all M>0 there exists N such that abs(xN)>M but i still don't understand how to show it.
 
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Here's something to try: Consider the following function of a real variable.

f(x)=[(x+1)/x]^3-x^3

Now take the derivative and show that f is monotonically decreasing for all x>0, and that the range of f is (0,infinity) when x>0. (sorry, LaTeX is acting funny)

What then can you conclude about f(n)?
 

What is the definition of unboundedness in mathematical terms?

Unboundedness refers to a sequence or set of numbers that has no upper or lower limit, meaning it can continue infinitely in either direction.

What is the formula for proving unboundedness of a sequence?

The formula for proving unboundedness of a sequence is [(n+1)/n]^3 - n^3. If this formula approaches infinity as n approaches infinity, then the sequence is unbounded.

What does it mean when a sequence is unbounded?

When a sequence is unbounded, it means that the values of the sequence can become arbitrarily large or small without any limit. This can also be thought of as the sequence having no upper or lower bound.

How can I prove that a sequence is unbounded using the formula [(n+1)/n]^3 - n^3?

To prove that a sequence is unbounded using this formula, you need to show that as n approaches infinity, the result of the formula approaches infinity as well. This can be done by taking the limit of the formula as n approaches infinity and showing that the result is infinity.

What are some real-world applications of proving unboundedness of a sequence?

Proving unboundedness of a sequence can be useful in various fields such as economics, physics, and computer science. For example, in economics, it can be used to model the growth of a population or the increase in prices over time. In physics, it can be used to describe the behavior of a system over time. In computer science, it can be used to analyze the performance of algorithms or the growth of data structures.

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