maths question about naming a harmonic function

I want to name a maths function because I will often mention it in my book I am writing but don't know what I should call it. I am using this function within certain discrete probability mass functions that use a special unique mathematical model which I have named the "progressive successor model of probability" and which only can work with a discrete variable which is a count of a series of events where each event may or may not have a successor -not sure if any of that info helps here.

I am aware of what is conventionally called the Riemann zeta function ζ(s) which outputs a list of mathematical constants where:

ζ(1) = ∑ [X = 1, ∞] 1 / ( X^1)
ζ(2) = ∑ [X = 1, ∞] 1 / ( X^2)
ζ(3) = ∑ [X = 1, ∞] 1 / ( X^3)
...etc

but what I want to know is if there is a conventional name for the similar function (or mathematical series ), which I will symbolize with f() just temporarily for now, which outputs a list of mathematical constants but where:

f(1) = ∑ [X = 1, ∞] 1 / (( X^1) (X + 1))
f(2) = ∑ [X = 1, ∞] 1 / (( X^2) (X + 1))
f(3) = ∑ [X = 1, ∞] 1 / (( X^3) (X + 1))
...etc.

?

and, if there is no such conventionally name for the above function then do any of you here, who generally know more about mathematics than I do, have any suggestions of what I could appropriately call it and what notation or letter or symbol I could appropriately use in my book to represent it?

+ Would it be exactly correct to call that function a "harmonic series" in my book or is it not quite literally a "harmonic series"?
I don't want to call it harmonic if that isn't strictly correct.

Originally posted by humy
I want to name a maths function because I will often mention it in my book I am writing but don't know what I should call it. I am using this function within certain discrete probability mass functions.

I am aware of what is conventionally called the Riemann zeta function ζ(s) which outputs a list of mathematical constants where:

ζ(1) = ∑ [X = 1, ∞] 1 ...[text shortened]... literally a "harmonic series"?
I don't want to call it harmonic if that isn't strictly correct.

Originally posted by DeepThought
Your function looks like it can be rewritten as a Dirichlet series, in which case this Wikipedia page might be of interest:

https://en.wikipedia.org/wiki/Dirichlet_series