I'm trying to find a general formula for a harmonic(ish) series

[al4n]

Im looking for a general formula for the partial sum of a series where the nth term is in the form of
a_n = 1/(c+kn),
where c and k are arbitrary constants.

it "looks" like a harmonic series but not in the form I'm capable of figuring out.
help.

 

[martinbn]

Do you mean the sum? It is divergent! Or do you mean the partial sums? Or something else?

 

[al4n]

Do you mean the sum? It is divergent! Or do you mean the partial sums? Or something else?

my bad I wasn't specific enough. I meant the partial sum.

 

[martinbn]

my bad I wasn't specific enough. I meant the partial sum.

What is that for the harmonic series?

 

[al4n]

What is that for the harmonic series?

what do you mean?

 

[martinbn]

what do you mean?

Well, you said.

it "looks" like a harmonic series but not in the form I'm capable of figuring out.
help.

It sounds like you can figure out the case of the harmonic series, and want to do the same with the more general. Is it not what you mean?

 

[al4n]

Well, you said.

It sounds like you can figure out the case of the harmonic series, and want to do the same with the more general. Is it not what you mean?

I guess. Is that not already figured out? Looking around, What I thought was the formula was in fact only an approximation. So what I should've first asked is, is there something like that to the specific example

 

[fresh_42]

I guess. Is that not already figured out? Looking around, What I thought was the formula was in fact only an approximation. So what I should've first asked is, is there something like that to the specific example

So to be clear: you are looking for a formula for $f(m;c,k)=\displaystyle{\sum_{n=1}^m}\dfrac{1}{c+kn}$ for any parameters $c,k \in \mathbb{R}$?

In that case, the answer is
https://www.wolframalpha.com/input?i=sum+(from+n=1+to+m)+1/(c+kn)=
https://en.wikipedia.org/wiki/Polygamma_function

 

[WWGD]

I'm not aware of the existence of a closed form for the ( Standard) Harmonic, only that the partial sums are never Integers. Uses Bertrand's lemma, which I believe it is now a theorem.