228.44630 41592 30813 25414 80861 26250 6

1st Solution
------------

Generalize the approach given in

    @article {MR528792,
    AUTHOR = {Baillie, Robert},
     TITLE = {Sums of reciprocals of integers missing a given digit},
   JOURNAL = {Amer. Math. Monthly},
  FJOURNAL = {The American Mathematical Monthly},
    VOLUME = {86},
      YEAR = {1979},
    NUMBER = {5},
     PAGES = {372--374},
      ISSN = {0002-9890},
     CODEN = {AMMYAE},
   MRCLASS = {40A05},
  MRNUMBER = {MR528792 (80g:40002)},
MRREVIEWER = {Joaquin Bustoz},
}

It gives 30D in about a second.

2nd Solution
------------

Dave Smith (Loyola Marymount University) reports the following:
(Jan 2, 2006)

I did it by generating a sequence of partial sums after k terms,
for k = 10, 10^2, ..., 10^10, then using repeated Aitken extra-
polation. So that is a very simple method, and gave about 11 s.d.

Actually, I started with a run in double precision and got 10^11
terms in 76 minutes. Using Kahan's summation trick gave all the
sums to more or less full precision, but starting with the third
repeated Aitken column the results seemed to have some instability
starting around the tenth s.d.  The last value in the Aitken^2
column was 228.4463041583602.

Then I ran it using multiple precision with 30 s.d. and got 10^10
terms in 13 hours 10 minutes.  Pretty good for software arithmetic.
Then Aitken was more stable, and the last value in the fourth Aitken
column was 228.4463041583748.

I stopped there, but if I wanted to try for another digit or two,
the sum would be easy to run in parallel on my dual processor 2.0Ghz
Mac.  Then I could get the multiple precision run up to 10^11 terms
in about 66 hours -- under 3 days.