find a particular solution of $A_{n} = A_{(n-1)} + f(n)$where $f(n)=n^{(c)}$,c is a random negative integer

Question

We all know that particular solution of $A_{n} = A_{(n-1)} + f(n)$

where $f(n)=n^c$ , c is a random positive integer.

Can be set to $(n^c+n^{(c-1)}+.....+1)$

But what about when $c\lt0$?

How do we find a particular solution of the form:

$A_{n} = A_{(n-1)} + f(n)$

where $f(n)=n^{(c)}$ , c is a random negative integer.

For example. What's the particular solution of $T(n)=T(n-1)+{1\over n}$

(P.S I know it's Harmonic series and we can use Integral test to prove that it's diverges by comparing its sum with an improper integral. But it doesn't matter. )

I can't find any clue in "Discrete Mathematics, 7th Edition".

Does anyone know the answer?

Answer 1

The solution is not trivial at all since $$a_n=\frac{(-1)^{c+1} \psi ^{-(c+1)}(n+1)}{(-c-1)!}+\text{Cte}$$ where appears the polygamma function

Answer 2

The $n$th partial sum of the Harmonic Series is called the $n$th Harmonic Number. According to Wikipedia, there's no closed form for the $n$th Harmonic Number.