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Does anyone know how to calculate this sum of finite series $$ S = \sum\limits_{k = 1}^n {\frac{1}{{ak + b}}} $$ with a and b are positive real numbers.

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Is that meant to be a partial sum from $k=1$ to $n$? It's not really an infinite series with the $n$ on top, did you mean to put $\infty$ on top? – Todd Wilcox Feb 1 '13 at 18:41
It's a finite series. Sorry for typos, my bad. – widapol Feb 1 '13 at 18:43
There is no elementary expression for that sum. You can find asymptotic expressions for it when $n\to\infty$, though. It will behave much like $\int_1^n(ax+b)^{-1}\,dx$ for large $n$, and that is an integral you can easily evaluate. – Harald Hanche-Olsen Feb 1 '13 at 18:53

You an write that as: $$ \frac{1}{a} \sum_{1 \le k \le n} \frac{1}{k + b / a} $$ If $\frac{b}{a}$ is an integer, you can express this in terms of harmonic numbers.

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You can write it as $(\Psi(b/a+n+1)-\Psi(b/a+1))/a$. See

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You can get the sum in terms of the $\psi$ function, here is a result by maple

$$ \frac{1}{a}\psi \left( n+1+{\frac {b}{a}} \right) -\frac{1}{a}\psi \left( 1+{\frac { b}{a}} \right). $$

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