Does the Limit of sin(n*alpha) as n Approaches Infinity Diverge?

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In summary: In this case, we are looking at the behavior of the sequence sin(n*alpha) as n goes to infinity, where alpha is a constant between 0 and pi. Since alpha is a fixed constant, we can use the same method as for a specific value of alpha, such as pi/3.In summary, we are trying to show that the sequence sin(n*alpha) does not have a limit as n goes to infinity for any value of alpha between 0 and pi.
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AndrejN96
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Homework Statement


Show that lim n->inf sin(n*alpha), 0 < alpha < pi, diverges.

Homework Equations


lim n-> inf sin(n) diverges

The Attempt at a Solution


I know how to solve this for a constant value of alpha (i.e pi/3), but am unaware of solving this one, where alpha may vary. I know that sin(alpha) when alpha is in the given interval has a positive value, but don't know how to apply that to this solution.
 
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If you can do it for specific values of alpha, you should be able to use the same method for any value of alpha (you say "am unaware of solving this one, where alpha may vary"- alpha is NOT varying in a specific sequence- it can be any number but is a fixed number).

By the way- you want to show that the sequence diverges, that it has NO limit, not that "the limit diverges".
 
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  • #3
AndrejN96 said:

Homework Statement


Show that lim n->inf sin(n*alpha), 0 < alpha < pi, diverges.

Homework Equations


lim n-> inf sin(n) diverges

The Attempt at a Solution


I know how to solve this for a constant value of alpha (i.e pi/3), but am unaware of solving this one, where alpha may vary. I know that sin(alpha) when alpha is in the given interval has a positive value, but don't know how to apply that to this solution.

As HallsofIvy has indicated, limits do not "converge" or "diverge"; they just either exist or do not exist. However, sequences can converge or diverge.
 

FAQ: Does the Limit of sin(n*alpha) as n Approaches Infinity Diverge?

What does it mean for a limit to diverge?

When a limit diverges, it means that the values of the function approach either positive or negative infinity as the input approaches a certain value.

How can you show that a limit diverges?

To show that a limit diverges, you can use various methods such as the squeeze theorem, the comparison test, or direct substitution to evaluate the limit. If the result is infinity, the limit diverges.

Can a limit diverge to a specific value?

No, a limit can only diverge to positive or negative infinity. It cannot converge to a specific value.

What is the difference between a limit diverging and a limit not existing?

A limit diverges when the function approaches infinity, while a limit does not exist when the function does not approach a specific value or approaches different values from the left and the right sides of the input.

Are there any special cases where a limit always diverges?

Yes, if a function has an infinite discontinuity at the input value, the limit will always diverge. This means that the function has a vertical asymptote at that point.

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