Does the Sequence a_n = sin(2πn) Converge or Diverge?

[Bipolarity]

Homework Statement

Does the following sequence converge, or diverge?
a_{n} = sin(2πn)

Homework Equations

The Attempt at a Solution

\lim_{n→∞} sin(2πn) does not exist, therefore the sequence should diverge? But it actually converges to 0?

I appreciate all help thanks.

BiP

 

[LCKurtz]

Write out the numerical value of a few terms of that sequence.

 

[Bipolarity]

Write out the numerical value of a few terms of that sequence.

They are all 0 ?? So the sequence converges, but how come when we apply the limit it diverges?

BiP

 

[HallsofIvy]

I have no idea what you mean by "when we apply the limit it diverges".

\lim_{n=0} sin(2\pi n)= 0, trivially.

 

[Bipolarity]

I have no idea what you mean by "when we apply the limit it diverges".

\lim_{n=0} sin(2\pi n)= 0, trivially.

I think there is some confusion between sin(2\pi n) defined for natural numbers, and that defined for real numbers. If we take the limit of it as it is defined for natural numbers, it converges to 0 trivially. But if we take the limit of it as it is defined for all real x, its limit does not exist.

Does this mean we cannot take the limit of the function defined for reals to determine the convergence of the function (or sequence) defined for naturals?

BiP

 

[LCKurtz]

I think there is some confusion between sin(2\pi n) defined for natural numbers, and that defined for real numbers. If we take the limit of it as it is defined for natural numbers, it converges to 0 trivially. But if we take the limit of it as it is defined for all real x, its limit does not exist.

Does this mean we cannot take the limit of the function defined for reals to determine the convergence of the function (or sequence) defined for naturals?

BiP

Yes and no. If the limit exists over the reals then it exists over the naturals. The converse is false. Also the limit may exist over the naturals but not over the reals as that example shows.

 

[Bipolarity]

Yes and no. If the limit exists over the reals then it exists over the naturals. The converse is false. Also the limit may exist over the naturals but not over the reals as that example shows.

If the limit over the reals diverges specifically to infinity, must the limit over the naturals also diverge to infinity?

BiP