Science Advisor
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- TL;DR
- About series where the complex phase of the terms rotates around the unit circle.
It's a basic fact that the harmonic series
##\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}##
diverges, while the similar series with alternating sign
##\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}##
converges to the value ##\log 2##.
If I compute partial sums for the series
##\displaystyle\sum_{n=1}^{\infty}\frac{e^{ic(n-1)}}{n}##
where ##c## is a real-valued constant, the real and imaginary parts seem to approach a finite value for any nonzero value of ##c##. The case ##c=\pi## is just the series with alternating sign.
Do these series with rotating phase factor appear in any applications? It seems to be a generalization of the concept whether a series of real-valued terms is alternating or has only terms of same sign.
##\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}##
diverges, while the similar series with alternating sign
##\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}##
converges to the value ##\log 2##.
If I compute partial sums for the series
##\displaystyle\sum_{n=1}^{\infty}\frac{e^{ic(n-1)}}{n}##
where ##c## is a real-valued constant, the real and imaginary parts seem to approach a finite value for any nonzero value of ##c##. The case ##c=\pi## is just the series with alternating sign.
Do these series with rotating phase factor appear in any applications? It seems to be a generalization of the concept whether a series of real-valued terms is alternating or has only terms of same sign.