example of summation by parts

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\sum_n \sin{nz}/n and \sum_n \cos{nz}/n

These series converge quite slowly and I don't know other ways to demonstrate their convergence than the one in "example of summation by parts". Are there other ways?

Re: \sum_n \sin{nz}/n and \sum_n \cos{nz}/n

Hmm, I might be totally wrong, but I seems like I always taken the convergence of these series for granted...

Isn't one of these series the Fourier series for the sawtooth blade function? The sawtooth blade function is of bounded variation, so Dirichlet's theorem applies.

Now, the proof of Dirichlet's theorem may use the second mean-value theorem for integrals, so I suppose the summation-by-parts argument is actually hidden there.

// Steve

Re: \sum_n \sin{nz}/n and \sum_n \cos{nz}/n

You mean the Dirichlet\'s convergence test? I have not known it, but now I found it in PM. Indeed, lieven uses the summation-by-parts in the proof.
Jussi

Re: \sum_n \sin{nz}/n and \sum_n \cos{nz}/n

> You mean the Dirichlet\'s convergence test? I have not known
> it, but now I found it in PM. Indeed, lieven uses the
> summation-by-parts in the proof.

No, I mean the famous Dirichlet theorem on convergence of Fourier series
(http://planetmath.org/encyclopedia/DirichletConditions.html)
which you surely must know?

Although that theorem you cite (which I also did not know previously)
does have a tangential connection.

// Steve

Re: \sum_n \sin{nz}/n and \sum_n \cos{nz}/n

> No, I mean the famous Dirichlet theorem on convergence of Fourier series (http://planetmath.org/encyclopedia/DirichletConditions.html)
which you surely must know?

Yes, I know it (although I didn't remember its name).

Jussi