Today I entered the following expression in maple: $$a_i = H_{10^i} - ln(10^i) - \gamma$$ Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant.

When I computed $a_n$ for $i = 0$ to $10$ I obtained the following results:

- $i = 0$; 4.227843350984671393934879099175975689578406640600764011942327651151323 * $10^{-1}$
- $i=1$; 4.9167496072675423629464709201487329610707429399557393414873118115813 * $10^{-2}$
- $i=2$; 4.991666749996032162622676207122311664609813510982102304110919767206 * $10^{-3}$
- $i=3;$ 4.999166666749999960317501984051226762153678825611388678758121701133 * $10^{-4}$
- $i=4$; 4.99991666666674999999960317460734126976551226762154503821179264423 * $10^{-5}$
- $i=5$; 4.9999916666666667499999999960317460321626984126226551226762154523 * $10^{-6}$
- $i=6$; 4.999999166666666666749999999999960317460317501984126984051226523 * $10^{-7}$
- $i=7$; 4.99999991666666666666674999999999999960317460317460734126984123 * $10^{-8}$
- $i=8$; 4.9999999916666666666666667499999999999999960317460317460321623 * $10^{-9}$
- $i=9$; 4.999999999166666666666666666749999999999999999960317460317423 * $10^{-10}$
- $i=10$; 4.99999999991666666666666666666674999999999999999999960317423 * $10^{-11}$

So we see that the periodic strips of ...99999..., of ...66666... and ...99999... an many other periods increase for even larger $i$. The question is now: Is there any rule behind it that the remainder term $a_i$ behaves that way?