## Complex Harmonic Progression

**Authors:** Jose R. Sousa

In $\textit{Generalized Harmonic Progression}$, we showed how to create formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, that is, $\sum_{j}1/(a j+b)^k$. Those formulae were more general than the ones we created in $\textit{Generalized Harmonic Numbers Revisited}$. In this new paper we make those formulae even more general by removing the restriction that $a$ and $b$ be integers, in other words, here we address $\sum_{j}1/(a\ii j+b)^k$, where $a$ and $b$ are complex numbers and $\ii$ is the imaginary unity. These new relatively simple formulae always hold, except when $\ii b/a\in \mathbb{Z}$. This paper employs a slightly modified version of the reasoning used previously. Nonetheless, we make another brief exposition of the principle used to derive such formulae.

**Comments:** 8 Pages. This paper derives a formula that holds for nearly all generalized harmonic progressions

**Download:** **PDF**

### Submission history

[v1] 2019-07-29 22:08:45

**Unique-IP document downloads:** 7 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*