Harmonic Series minus terms with “9” Converges!

Something interesting I saw on: http://blogs.scientificamerican.com/roots-of-unity/what-the-prime-number-tweetbot-taught-me-about-infinite-sums/?WT.mc_id=SA_WR_20160504

The harmonic series is the infinite sum 1+1/2+1/3+1/4+…, the sum of the reciprocals of every positive integer. Even though the terms get closer and closer to 0, the series goes to infinity, meaning it eventuallygets bigger than any number you can throw at it. (The question is fun to think about, so I won’t spoil it for you, but Wikipedia has an explanation.)

It seems baffling that if we take away 1/9, 1/19, 1/29, and so on, the series converges instead. In fact, it’s less than 90, which is frankly pitiful compared to the infinitude of the harmonic series.