Proving Infinity of a Sequence: a(n)=[n+7]/[2+sin(n)]

In summary, the sequence (a(n)) can be proven to tend to infinity as n tends to infinity by using the fact that 1 <= 2 + sin(n) <= 3 and showing that the lower and upper limits of (n + 7)/(2 + sin(n)) both have the same limit as n gets large. This technique is more effective than using L'Hopital's Rule, which does not apply in this case.
  • #1
kmeado07
40
0

Homework Statement



Prove that the following sequence (a(n)) has the property that a(n) tends to infinity as n tends to infinity.


Homework Equations


a(n)=[n+7]/[2+sin(n)]


The Attempt at a Solution



i tried l'hopitals rule, so i got 1/cos(n)...which wouldn't work.
so I am not sure how to do this question.
 
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  • #2


L'Hopital's Rule doesn't apply here. The quotient has to be of the form 0/0 or infinity/infinity for L'H's Rule to be used.

Instead, use the fact that 1 <= 2 + sin(n) <= 3, for all n to get lower and upper limits on the value of (n + 7)/(2 + sin(n)), and then show that the lower and upper limit expressions both have the same limit as n gets large, thereby trapping the expression in the middle.
 
  • #3


Since n+7 is unbounded, to apply l'Hopital to this you would need that the denominator is also unbounded (so you have an infinity/infinity form). It's not. The denominator is bounded and positive. That should tell you the limit right there.
 

1. What is a sequence infinity proof?

A sequence infinity proof is a mathematical proof that demonstrates that a sequence of numbers will continue to increase or decrease indefinitely, without reaching a specific value or limit.

2. How is a sequence infinity proof different from other types of mathematical proofs?

A sequence infinity proof is specific to sequences of numbers, while other types of mathematical proofs may be used to prove a variety of mathematical statements or theorems.

3. What are the key components of a sequence infinity proof?

The key components of a sequence infinity proof include showing that the sequence is increasing or decreasing, providing an expression for the general term of the sequence, and demonstrating that the sequence will continue to increase or decrease without bound.

4. Why are sequence infinity proofs important in mathematics?

Sequence infinity proofs are important in mathematics because they provide a rigorous way to prove that a sequence will continue to increase or decrease indefinitely, which can have important implications in various fields of mathematics and science.

5. Can a sequence have more than one infinity proof?

Yes, a sequence can have multiple infinity proofs depending on the specific properties and characteristics of the sequence. Different approaches and techniques may be used to prove the infinity of a sequence, leading to multiple proofs for the same sequence.

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