On 18/12/2017 22:46, Dan Asimov wrote:
I recently tried to make a plot of
(*) y = Sum_{1 <= n < oo} sin(nx)
for some small values of infinity, and it distinctly hovered around the apparent value
Sum_{1 <= n < oo} Im( exp(inx))
= Im(Sum_{1 <= n < oo} exp(inx))
=(???) Im( exp(ix) / (1 - exp(ix)) ) = sin(x) / (2 - 2*cos(x)).
Clearly the original series converges only for x = n*pi for some integer n.
Can this function
f(x) = sin(x) / (2 - 2*cos(x))
be the *Cesaro sum* of the original series (*) ???
Or at least for certain values of x ???
This seems highly relevant: https://projecteuclid.org/download/pdf_1/euclid.bams/1183504486 (Further remark: f(x) is certainly the *Abel* sum of the series, and any Cesaro-summable series is Abel-summable with the same sum.) -- g