Sin(npi/2) non-convergence to 0

[peripatein]

Hello,

I'd like to show that sin(npi/2) does not converge. I have managed to show that it does not converge to +-1, but how may I also prove that it does not converge to 0?

 

[Dick]

sin(0*pi/2)=0, sin(1*pi/2)=1, sin(2*pi/2)=0, sin(3*pi/2)=(-1), sin(4*pi/2)=0. Now just kind of extrapolate.

 

[peripatein]

I have to demonstrate it using the non-existence of limit definition. Hence, using epsilon etc.

 

[Dick]

I have to demonstrate it using the non-existence of limit definition. Hence, using epsilon etc.

How did you show it doesn't converge to +/-1? Why is it harder to show it doesn't converge to 0?

 

[micromass]

I have to demonstrate it using the non-existence of limit definition. Hence, using epsilon etc.

Sure. But what Dick is suggesting is that you actually calculate $\sin(n\pi /2)$ and to describe the sequence without sines. This makes it way easier to prove that a limit does not exist.

 

[peripatein]

In order to prove that it does not converge to +-1, I simply defined it as (-1)^n and then showed that for any even n (or odd n, in the case of limit being +1) |(-1)^n - 1| >= epsilon (taking epsilon=1).
But what would be the general term, the equivalent of (-1)^n, for 0? I mean, I'd still have to prove that the sequence does not converge to 0 in order for my proof to be complete.

 

[Dick]

In order to prove that it does not converge to +-1, I simply defined it as (-1)^n and then showed that for any even n (or odd n, in the case of limit being +1) |(-1)^n - 1| >= epsilon (taking epsilon=1).
But what would be the general term, the equivalent of (-1)^n, for 0? I mean, I'd still have to prove that the sequence does not converge to 0 in order for my proof to be complete.

Now why would you 'define' sin(pi*n/2) to be (-1)^n when it isn't true? That would seem like a losing strategy for proving something?

 

[peripatein]

It is true and is equal to (-1)^n for all odd positions, i.e. n=2k+1.

 

[Dick]

It is true and is equal to (-1)^n for all odd positions, i.e. n=2k+1.

I'd write that as (-1)^k but alright. Your sequence is 0 for even n. It's also true |(-1)^k-0|>=1. Can't you use the same argument? There's no rule against comparing even terms with odd terms.