I am tempted to think that the Mathematica output is actually correct, and the erroneous numerical result for small $a$ is a numerical instability in the computation of the digamma function. See this plot that compares the digamma expression (blue) with a numerical evaluation of the sum (gold), as a function of $a$. For $a\gtrsim 0.01$ the two answers line up, and the deviations for smaller $a$ are widely oscillating, indicating a numerical instability.
