What do we know about the series $\sum_{k=1}^\infty \frac{1}{x+k}$? If possible, I would like to find a closed-form expression for
$$
f(x)=\sum_{k=1}^\infty \frac{1}{x+k}
$$
The series is so simple (possibly deceptively so) that I'm sure it has been studied before somewhere, but I don't know what to call it and where to find it.
What is this series called so I can learn more? I know I can expand it into a double series by expanding $\frac{1}{1-x/k}$ for $|x|<k$ and $\frac{1}{1-k/x}$ for $|x|>k$ and working on the partial series, which I did before in the special case of $|x|<1$, and the result yields a sum over Bernoulli numbers that I'm not familiar with.
Is a nice closed-form solution for this series known?
 A: This series is also basically a part of the Hurwitz zeta function in the so-called ``s=1'' case. It was introduced in the later half of the 1800's and comes up in analytic number theory. There is no closed form and the theory is not elementary.
A: Consider $$f_n(x)=\sum_{k=1}^n \frac{1}{x+k}=\psi(n+x+1)-\psi (x+1)=H_{n+x}-H_x$$ where appear the digamma function and the genarlized harmonic numbers.
Use the asymptotics
$$H_p=\gamma +\log (p)+\frac{1}{2 p}-\frac{1}{12
   p^2}+O\left(\frac{1}{p^4}\right)$$ and make $p=n+x$. Then continuing with Taylor series
$$f_n(x)=\log(n)-\psi
   (x+1)+\frac{2x+1}{2n}-\frac{6 x^2+6 x+1}{12 n^2}+O\left(\frac{1}{n^3}\right)$$
A: $$
\frac 1 {\lfloor x \rfloor + k+1} \le \frac 1 {x+k} \le \frac 1 {\lfloor x \rfloor + k}
$$
The two series whose terms are the first and the third expressions above are just tail ends of the harmonic series that diverges to $+\infty.$
A: Using the following definitions
$$
\psi(z) = -\gamma + \sum_{n=0}^\infty \left( \frac 1 {n+1} - \frac 1 {n+z} \right)
$$
$$
\gamma = \lim_{n\,\to\,\infty} \left( -\ln n + \sum_{k=1}^n \frac 1 k \right)
$$
one can conclude that
\begin{align}
& \sum_{k=1}^\infty \frac{1}{x+k}=\sum_{k=1}^\infty \frac{1}{k+1}-\gamma-\psi(x)+\frac{x-1}{x} \\[6pt]
= {} & \lim_{n \to \infty}  \ln(n)-\psi(x)+\frac{3x-2}{2x}.
\end{align}
