Pointwise convergence for all real x

[lys04]

Homework Statement

pointwise convergence of a series

Relevant Equations

$$ \sum_{n=1}^\infty sin(\frac{n\pi}{2})sin(nx) $$

How do I know whether or not the series
$$ \sum_{n=1}^\infty sin(\frac{n\pi}{2})sin(nx)$$

converges pointwise for all real x or not?

By the way am I right in thinking that converging pointwise for all real x means whatever x i plug into the series it converges to some finite value?

I was thinking if i plug in x=pi/2 then I'd get

$$ \sum_{n=1}^\infty sin^2(\frac{n\pi}{2}) $$

Which diverges, does that prove that the series doesn't converge pointwise for all x?

 

[Orodruin]

How do I know whether or not the series
$$ \sum_{n=1}^\infty sin(\frac{n\pi}{2})sin(nx)$$

converges pointwise for all real x or not?

You either show that it does or find a way to show that it does not.

By the way am I right in thinking that converging pointwise for all real x means whatever x i plug into the series it converges to some finite value?

Yes.

I was thinking if i plug in x=pi/2 then I'd get

$$ \sum_{n=1}^\infty sin^2(\frac{n\pi}{2}) $$

Which diverges, does that prove that the series doesn't converge pointwise for all x?

Yes.