Some convergent subseries of the harmonic series

The harmonic series diverges. But, Kempner [A curious convergent series, Amer. Math Monthly 21 (1914) 48–50] proved that by removing those terms from the harmonic series whose denominators contain a digit [Formula: see text] anywhere, the resulting sub-series converges. Thus, Kempner allowed no 9’s at all, but in this short note, we allow 9’s with some restrictions and without tampering the convergence of the resultant series. Some results of our note generalize a recent note of Mukherjee and Sarkar [A short note on a curious convergent series, Asian-Eur. J. Math. 14(09) (2021) 2150158, doi:10.1142/S1793557121501588 ].