The harmonic series Σ(1/n) diverges. What if you remove all terms with the digit 9 in the denominator? Does it still diverge? (Spoiler: the next tweet contains the answer.) [1/4]

The sum converges. It is not surprising if you think about it the right way—most large numbers have every digit, so you are throwing away a lot of terms from the harmonic series. In particular, of the 9*10^{d-1} numbers with d digits, 8*9^{d-1} have no 9. Each of these [2/4]

d-digit numbers is at least 10^{d-1}. Using the comparison test, the sum of the series is less than
Σ(8*[9/10]^{d-1})=80. So it converges. Thanks to Wikipedia, I now know that this series is called the Kempner series and that the sum is approximately 22.92. [3/4]

This tweet was inspired by @MrHonner who tweeted about the Harmonic series earlier today. [4/4]