# A generalization of alternating series involving modulus?

Alternating series are common in the literature, with important examples including

$\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n}=\log 2$,

$\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n^2}=\frac{\pi^2}{12}$,

both specializations of

$\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n^s}=\eta(s)=(1-2^{1-s})\zeta(s)$

where $\eta(s)$ is the Dirichlet eta function, and $\zeta(s)$ is the Riemann Zeta function. Many, many Taylor series are also alternating, too, needless to say.

Recently, I've been exploring the following alternating series generalization.

Define $m_a(n)=n(\mod a)-(n-1)(\mod a)$.

Thus, $m_2(n)$ cycles through $1,-1,1,-1,...$, $m_3(n)$ cycles through $1,1,-2,1,1,-2,...$, $m_4(n)$ cycles through $1,1,1,-3,1,1,1,-3,...$, and so on, with the cycle consisting of $a$ elements before repeating.

$m_2(n)$ equals $(-1)^{n-1}$, suggesting $m_a(n)$ is one way to generalize alternating series. And, at least empirically,

$\displaystyle\sum_{n=1}\frac{m_a(n)}{n}=\log a$,

$\displaystyle\sum_{n=1}\frac{m_a(n)}{n^2}=\frac{\pi^2}{6} - \frac{\pi^2}{6a}$,

and

$\displaystyle\sum_{n=1}\frac{m_a(n)}{n^s}=(1-a^{1-s})\zeta(s)$

My questions:

1) Does this sort of generalization of alternating series have a name? Is there a larger body of work elaborating the use of such approaches that I could read more about?

2) Here, $m_a(n)$ only supports $a$ as a positive integer > 1; is there a way to generalize this further to support $a$ as a negative, real, or even complex variable? Or is there a way to prove that there isn't?

3) From further experiments, not all alternating series seem to produce interesting results when generalized from $(-1)^{n-1}$ to $m_a(n)$. Is there a way to tell, a priori, which sorts of alternating series are fruitful to examine with it?

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I think the natural generalization is to functions $m(n)$ periodic with some period $k$ and with $\sum_1^km(n)=0$. – Gerry Myerson Oct 23 '13 at 22:17