Is Sequence xn Unbounded? Quickest Solution

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In summary, the sequence xn=[(n5+7n+3)7]/[(7-n4)6] is not bounded because as n tends to ∞, xn also tends to ∞. This can be determined by looking at the highest degree of n in the numerator and denominator, which is 35 and 24 respectively.
  • #1
gottfried
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Homework Statement


Is the sequence xn=[(n5+7n+3)7]/[(7-n4)6] bounded?


]



The Attempt at a Solution



I've managed to tell that the sequence is not bound because as n tends to ∞ xn also tends to ∞ but it took me a relatively long time.
Is there any way of telling this by just looking at the exponents or what is the quickiest way to tell that this sequence will get infinitely large.
 
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  • #2
gottfried said:

Homework Statement


Is the sequence xn=[(n5+7n+3)7]/[(7-n4)6] bounded?


]



The Attempt at a Solution



I've managed to tell that the sequence is not bound because as n tends to ∞ xn also tends to ∞ but it took me a relatively long time.
Is there any way of telling this by just looking at the exponents or what is the quickiest way to tell that this sequence will get infinitely large.

The highest degree of n in the numerator is 35 and in the denominator is 24. That's all you need to know.
 
  • #3
I was pretty sure it was something simple. Thanks!
 

1. What does it mean for a sequence to be unbounded?

A sequence is unbounded if its terms do not have a finite limit, meaning that they continue to increase or decrease without bound.

2. How can I determine if a sequence is unbounded?

To determine if a sequence is unbounded, you can examine the behavior of its terms as they approach infinity. If the terms continue to increase or decrease without bound, then the sequence is unbounded.

3. Why is it important to know if a sequence is unbounded?

Knowing if a sequence is unbounded can help in understanding the behavior and limits of a mathematical function. It can also help in making predictions and solving problems in various fields such as physics, engineering, and economics.

4. Is there a quickest solution to determine if a sequence is unbounded?

Yes, there is a quick solution to determine if a sequence is unbounded. You can use the Monotone Convergence Theorem, which states that a sequence is unbounded if and only if it is not bounded. This means that if you can prove that the terms in a sequence are either always increasing or always decreasing, then the sequence is unbounded.

5. Can a sequence be both bounded and unbounded?

No, a sequence cannot be both bounded and unbounded. A sequence can only have one of these properties. If a sequence is bounded, it means that its terms have a finite limit or are restricted to a certain range. On the other hand, an unbounded sequence has terms that continue to increase or decrease without bound.

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